Number Theory — CAT Previous-Year Questions with Solutions

CAT 2024 Slot 1 · QA

Q1.

For any natural number n, let a_n be the largest integer not exceeding √n. Then the value of a₁ + a₂ + a₃ + ... + a₅₀ is

CAT 2024 Slot 1 · QA

Q2.

The sum of all four-digit numbers that can be formed with the distinct non-zero digits a, b, c and d, with each digit appearing exactly once in every number, is 153310 + n, where n is a single digit natural number. Then, the value of (a + b + c + d + n) is

CAT 2024 Slot 1 · QA

Q3.

Let x, y, and z be real numbers satisfying

4(x² + y² + z²) = a,

4(x - y - z) = 3 + a

Then a equals

CAT 2024 Slot 1 · QA

Q4.

When 10¹⁰⁰ is divided by 7, the remainder is

CAT 2024 Slot 2 · QA

Q5.

If x and y are real numbers such that 4x² + 4y² - 4xy - 6y + 3 = 0, then the value of (4x + 5y) is

CAT 2024 Slot 2 · QA

Q6.

When 3³³³ is divided by 11, the remainder is

CAT 2024 Slot 3 · QA

Q7.

If 10⁶⁸ is divided by 13, the remainder is

CAT 2023 Slot 1 · QA

Q8.

If x and y are real numbers such that x² + (x – 2y - 1)² = -4y(x + y), then the value of x - 2y is?

CAT 2023 Slot 1 · QA

Q9.

Let n be the least positive integer such that 168 is a factor of 1134ⁿ. If m is the least positive integer such that 1134ⁿ is a factor of 168^m, then m + n equals

CAT 2023 Slot 2 · QA

Q10.

For any natural numbers m, n and k, such that k divides both m + 2n and 3m + 4n, k must be a common divisor of

CAT 2023 Slot 2 · QA

Q11.

The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is

CAT 2023 Slot 2 · QA

Q12.

If p² + q² - 29 = 2pq - 20 = 52 - 2pq, then the difference between the maximum and minimum possible value of (p³ - q³) is

CAT 2023 Slot 3 · QA

Q13.

For some real numbers a and b, the system of equations x + y = 4 and (a + 5)x + (b² -15)y = 8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is?

CAT 2023 Slot 3 · QA

Q14.

Let n and m be two positive integers such that there are exactly 41 integers greater than 8^m and less than 8ⁿ, which can be expressed as powers of 2. Then, the smallest possible value of n + m is?

CAT 2023 Slot 3 · QA

Q15.

The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

CAT 2022 Slot 1 · QA

Q16.

Let a and b be natural numbers. If a² + ab + a = 14 and b² + ab + b = 28, then (2a + b) equals

CAT 2022 Slot 1 · QA

Q17.

Let A be the largest positive integer that divides all the numbers of the form 3^k + 4^k + 5^k and B be the largest positive integer that divides all the numbesr of the form 4^k + 3(4^k) + 4^k+2, where k is any positive integer. Then (A + B) equals

CAT 2022 Slot 1 · QA

Q18.

For natural numbers x, y and z, if xy + yz = 19 and yz + xz = 51, then the minimum possible value of xyz is

CAT 2022 Slot 2 · QA

Q19.

If a and b are non-negative real numbers such that a + 2b = 6, then the average of the maximum and minimum values of (a + b) is:

CAT 2022 Slot 2 · QA

Q20.

For some natural number n, assume that (15,000)! is divisible by (n!)!. The largest possible value of n is:

CAT 2022 Slot 3 · QA

Q21.

A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the students in the school is

CAT 2021 Slot 2 · QA

Q22.

For a 4-digit number, the sum of its digits in the thousands, hundreds and tens places is 14, the sum of its digits in the hundreds, tens and units places is 15, and the tens place digit is 4 more than the units place digit. Then the highest possible 4-digit number satisfying the above conditions is

CAT 2021 Slot 2 · QA

Q23.

Consider the pair of equations: x² – xy – x = 22 and y² – xy + y = 34. If x > y, then x – y equal.

CAT 2021 Slot 3 · QA

Q24.

The cost of fencing a rectangular plot is ₹ 200 per ft along one side, and ₹ 100 per ft along the three other sides. If the area of the rectangular plot is 60000 sq. ft, then the lowest possible cost of fencing all four sides, in INR, is

CAT 2020 Slot 1 · QA

Q25.

How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?

CAT 2020 Slot 1 · QA

Q26.

If a, b and c are positive integers such that ab = 432, bc = 96 and c < 9, then the smallest possible value of a + b + c is

CAT 2020 Slot 2 · QA

Q27.

For real x, the maximum possible value of $\frac{x}{\sqrt{1 + x^{4}}}$ is

CAT 2020 Slot 2 · QA

Q28.

If x and y are non-negative integers such that x + 9 = z, y + 1 = z and x + y < z + 5, then the maximum possible value of 2x + y equals

CAT 2020 Slot 2 · QA

Q29.

If x and y are positive real numbers satisfying x + y = 102, then the minimum possible value of 2601 $(1 + \frac{1}{x}) (1 + \frac{1}{y})$ is

CAT 2020 Slot 2 · QA

Q30.

The number of pairs of integer (x, y) satisfying x ≥ y ≥ - 20 and 2x + 5y = 99 is

CAT 2020 Slot 3 · QA

Q31.

Let m and n be natural numbers such that n is even and 0.2 < $\frac{m}{20}, \frac{n}{m}, \frac{n}{11}$ < 0..5. Then m – 2n equals

CAT 2020 Slot 3 · QA

Q32.

Let N, x and y be positive integers such that N = x + y, 2 < x < 10 and 14 < y < 23. If N > 25, then how many distinct values are possible for N?

CAT 2020 Slot 3 · QA

Q33.

How many of the integers 1, 2, … , 120, are divisible by none of 2, 5 and 7?

CAT 2019 Slot 1 · QA

Q34.

The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157 : 3, then the sum of the two numbers is

CAT 2019 Slot 2 · QA

Q35.

Let a, b, x, y be real numbers such that a² + b² = 25 , x² + y² = 169 and ax + by = 65. If k = ay - bx, then

CAT 2019 Slot 2 · QA

Q36.

How many factors of 2⁴ × 3⁵ × 10⁴ are perfect squares which are greater than 1?

CAT 2019 Slot 2 · QA

Q37.

What is the largest positive integer such that $\frac{(n^{2} + 7 n + 12)}{(n^{2} - n - 12)}$ is also positive integer?

CAT 2019 Slot 2 · QA

Q38.

How many pairs (m,n) of positive integers satisfy the equation m² + 105 = n²?

CAT 2018 Slot 1 · QA

Q39.

The number of integers x such that 0.25 ≤ 2^x ≤ 200, and 2^x + 2 is perfectly divisible by either 3 or 4, is

CAT 2018 Slot 1 · QA

Q40.

While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is

CAT 2018 Slot 2 · QA

Q41.

How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?

CAT 2018 Slot 2 · QA

Q42.

If A = {6²ⁿ - 35n - 1: n = 1, 2, 3, ...} and B = {35(n - 1) : n = 1,2,3,...} then which of the following is true?

CAT 2017 Slot 1 · QA

Q43.

If x + 1 = x² and x > 0, then 2x⁴ is:

CAT 2017 Slot 2 · QA

Q44.

If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is

CAT 2017 Slot 2 · QA

Q45.

How many different pairs (a, b) of positive integers are there such that a ≤ b and $\frac{1}{a} + \frac{1}{b} = \frac{1}{9}$ ?

CAT 2008 · QA

Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Q46.

Suppose, the seed of any positive integer n is defined as follows:

seed(n) = n, if n < 10
= seed(s(n)), otherwise,

where s(n) indicates the sum of digits of n.

For example, seed(7) = 7, seed(248) = seed(2 + 4 + 8) = seed(14) = seed(1 + 4) = seed(5) = 5 etc.

How many positive integers n, such that n < 500, will have seed(n) = 9?

CAT 2008 · QA

Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Q47.

What are the last two digits of 7²⁰⁰⁸?

CAT 2008 · QA

Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Q48.

Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is a perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say m, of these three integers?

CAT 2007 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Cities A and B are in different time zones. A is located 3000 km east of B. The table below describes the schedule of an airline operating non-stop flights between A and B. All the times indicated are local and on the same day.

Assume that planes cruise at the same speed in both directions. However, the effective speed is influenced by a steady wind blowing from east to west at 50 km per hour.

ââââââââââââââ

Q49.

Consider all four digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?

CAT 2007 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Let a₁ = p and b₁ = q, where p and q are positive quantities.

Define:
a_n = pb_n−1 b_n = qb_n−1, for even n > 1 and
a_n = pa_{n − 1} b_n = qa_{n − 1}, for odd n > 1.

Q50.

Each question is followed by two statements A and B. Answer each question using the following instructions.
Mark (1) if the question can be answered by using statement A alone but not by using statement B alone.
Mark (2) if the question can be answered by using statement B alone but not by using statement A alone.
Mark (3) if the question can be answered by using both the statements together but not by using either of the statements alone.
Mark (4) if the question cannot be answered on the basis of the two statements.

Consider integers x, y and z. What is the minimum possible value of x² + y² + z² ?

A. x + y + z = 89
B. Among x, y, z two are equal.

CAT 2007 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Let a₁ = p and b₁ = q, where p and q are positive quantities.

Define:
a_n = pb_n−1 b_n = qb_n−1, for even n > 1 and
a_n = pa_{n − 1} b_n = qa_{n − 1}, for odd n > 1.

Q51.

How many pairs of positive integers m, n satisfy 1/m + 4/n = 1/12 where n is an odd integer less than 60?

CAT 2006 · QA

Q52.

The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can possibly be one of these four numbers?

CAT 2006 · QA

Passage / Data

Answer the following question based on the information given below.

An airline has a certain free luggage allowance and charges for excess luggage at a fixed rate per kg. Two passengers, Raja and Praja have 60 kg of luggage between them, and are charged Rs. 1200 and Rs. 2400 respectively for excess luggage. Had the entire luggage belonged to one of them, the excess luggage charge would have been Rs. 5400.

Q53.

The number of employees in Obelix Menhir Co. is a prime number and is less than 300. The ratio of the number of employees who are graduates and above, to that of employees who are not, can possibly be:

CAT 2005 · QA

Q54.

If x = (16³ + 17³ + 18³ + 19³), then x divided by 70 leaves a remainder of

CAT 2005 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Q55.

If R = $\frac{30^{65} - 29^{65}}{30^{64} + 29^{64}}$ then

CAT 2005 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Q56.

Let n! = 1 × 2 × 3 × ... × n for integer n ≥ 1. If p = 1! + (2 × 2!) + (3 × 3!) + … + (10 × 10!), then p + 2 when divided by 11! leaves a remainder of

CAT 2005 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Q57.

The digits of a three-digit number A are written in the reverse order to form another three-digit number B. If B > A and B − A is perfectly divisible by 7, then which of the following is necessarily true?

CAT 2005 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Q58.

The rightmost non-zero digit of the number 30²⁷²⁰ is

CAT 2005 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Q59.

For a positive integer n, let P_n denote the product of the digits of n, and S_n denote the sum of the digits of n. The number of integers between 10 and 1000 for which P_n + S_n = n is

CAT 2005 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Q60.

Let S be a set of positive integers such that every element n of S satisfies the conditions

a. 1000 ≤ n ≤ 1200
b. every digit in n is odd

Then how many elements of S are divisible by 3?

CAT 2004 · QA

Q61.

Suppose n is an integer such that the sum of the digits of n is 2, and 10¹⁰ < n < 10¹¹. The number of different values for n is

CAT 2004 · QA

Passage / Data

Answer the following question based on the information given below.

In an examination, there are 100 questions divided into three groups A, B and C such that each group contains at least one question. Each question in group A carries 1 mark, each question in group B carries 2 marks and each question in group C carries 3 marks. It is known that the questions in group A together carry at least 60% of the total marks.

Q62.

The remainder, when (15²³ + 23²³) is divided by 19, is:

CAT 2003 Slot 1 · QA

Passage / Data

Answer the following question based on the information given below.

A city has two perfectly circular and concentric ring roads, the outer ring road (OR) being twice as long as the inner ring road (IR). There are also four (straight line) chord roads from E1, the east end point of OR to N2, the north end point of IR; from N1, the north end point of OR to W2, the west end point of IR; from W1, the west end point of OR, to S2, the south end point of IR; and from S1, the south end point of OR to E2, the east endpoint of IR. Traffic moves at a constant speed of 30π km/hr on the OR road, 20π km/hr on the IR road, and km/hr on all the chord roads.

Q63.

How many even integers n, where 100 ≤ n ≤ 200, are divisible neither by seven nor by nine?

CAT 2003 Slot 1 · QA

Passage / Data

Answer the following question based on the information given below.

A city has two perfectly circular and concentric ring roads, the outer ring road (OR) being twice as long as the inner ring road (IR). There are also four (straight line) chord roads from E1, the east end point of OR to N2, the north end point of IR; from N1, the north end point of OR to W2, the west end point of IR; from W1, the west end point of OR, to S2, the south end point of IR; and from S1, the south end point of OR to E2, the east endpoint of IR. Traffic moves at a constant speed of 30π km/hr on the OR road, 20π km/hr on the IR road, and km/hr on all the chord roads.

Q64.

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

Is a⁴⁴ < b¹¹, given that a = 2 and b is an integer?

b is even
b is greater than 16

CAT 2003 Slot 1 · QA

Passage / Data

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

Q65.

The number of positive integers n in the range 12 ≤ n ≤ 40 such that the product (n − 1)(n − 2)...3.2.1 is not divisible by n is

CAT 2003 Slot 1 · QA

Passage / Data

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

Q66.

Let T be the set of integers {3, 11, 19, 27, ..., 451, 459, 467} and S be a subset of T such that the sum of no two elements of S is 470. The maximum possible number of elements in S is

CAT 2003 Slot 2 · QA

Q67.

If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is:

CAT 2003 Slot 2 · QA

Q68.

Let x and y be positive integers such that x is prime and y is composite. Then,

CAT 2003 Slot 2 · QA

Q69.

If n is such that 36 ≤ n ≤ 72, then x = $\frac{n^{2} + 2 \sqrt{n} (n + 4) + 16}{n + 4 \sqrt{n} + 4}$ satisfies

CAT 2003 Slot 2 · QA

Q70.

If 13x + 1 < 2z, and z + 3 = 5y², then

CAT 2003 Slot 2 · QA

Q71.

Let n(>1) be a composite integer such that $\sqrt{n}$ is not an integer.

Consider the following statements:

A: n has a perfect integer - valued divisor which is greater than 1 and less than $\sqrt{n}$ .

B: n has a perfect integer- valued divisor which is greater than $\sqrt{n}$ but less than n.

Then,

CAT 2003 Slot 2 · QA

Passage / Data

Answer the following question based on the information given below.

Consider three circular parks of equal size with centres at A₁, A₂ and A₃ respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles A₁A₂A₃, B₁B₂B₃ and C₁C₂C₃, as shown. Three sprinters A, B, and C begin running from points A₁, B₁ and C₁ respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.

Q72.

What is the remainder when 4⁹⁶ is divided by 6?

CAT 2003 Slot 2 · QA

Passage / Data

Answer the following question based on the information given below.

A string of three English letters is formed as per the following rules:

The first letter is any vowel.
The second letter is m, n or p.
If the second letter is m, then the third letter is any vowel which is different from the first letter.
If the second letter is n, then the third letter is e or u.
1. If the second letter is p, then the third letter is the same as the first letter.

Q73.

If three positive real numbers x, y, z satisfy y – x = z – y and xyz = 4, then what is the minimum possible value of y?

CAT 2003 Slot 2 · QA

Passage / Data

Answer the following question based on the information given below.

A string of three English letters is formed as per the following rules:

The first letter is any vowel.
The second letter is m, n or p.
If the second letter is m, then the third letter is any vowel which is different from the first letter.
If the second letter is n, then the third letter is e or u.
1. If the second letter is p, then the third letter is the same as the first letter.

Q74.

What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7?

CAT 2002 · QA

Q75.

On dividing a number by 3, 4 and 7, the remainders are 2, 1 and 4 respectively. If the same number is divided by 84 then the remainder is

CAT 2002 · QA

Q76.

There are three pieces of cake weighing 9/2 lbs, 27/4 lbs and 36/5 lbs. Pieces of the cake are equally divided and distributed in such a manner that every guest in the party gets one single piece of cake. Further the weight of the pieces of the cake is as heavy as possible. What is the largest number of guests to whom we can distribute the cake?

CAT 2002 · QA

Q77.

For three real numbers x, y and z, x + y + z = 5, and xy + yz + xz = 3. What is the largest value which x can take?

CAT 2002 · QA

Q78.

If x² + 5y² + z² = 2y(2x + z), then which of the following statements are necessarily true?

I. x = 2y
II. x = 2z
III. 2x = z

CAT 2002 · QA

Q79.

Number S is equal to the square of the sum of the digits of a 2 digit number D. If the difference between S and D is 27, then D is

CAT 2002 · QA

Q80.

For all integers n > 0, 7⁶ⁿ – 6⁶ⁿ is divisible by

CAT 2002 · QA

Q81.

If U, V, W and m are natural numbers such that U^m + V^m = W^m, then which of the following is true?

CAT 2002 · QA

Q82.

The remainder when 2²⁵⁶ is divided by 17 is

CAT 2002 · QA

Passage / Data

Sum of first n natural numbers = S(n)

Sum given by student = 575

S(10) = $\frac{10 \times 11}{2} =$ 55

S(20) = $\frac{20 \times 21}{2} =$ 210

S(30) = $\frac{30 \times 31}{2} =$ 465

S(40) = $\frac{40 \times 41}{2} =$ 820

∴ The student stopped counting somewhere between 30 and 40.

Consider S(35) = $\frac{36 \times 35}{2} =$ 630

The student stopped somewhere before 35.

∴ S(31) = 496, S(32) = 528, S(33) = 561 and S(34) = 595

But the student gave 575 as the sum, so the student missed on the number 20.

Hence, option 4.

Q83.

A rich merchant had collected many gold coins. He did not want anybody to know about them. One day, his wife asked. "How many gold coins do we have?" After pausing a moment, he replied, "Well! If I divide the coins into two unequal numbers, then 48 times the difference of the numbers is equal to the difference of their squares. The wife looked puzzled. Can you help the merchant's wife by finding out how many gold coins the merchant has?

CAT 2002 · QA

Passage / Data

Answer the following question based on the information given below.

There are 11 alphabets A, H, I, M, O, T, U, V, W, X, Y. They are called symmetrical alphabets. The remaining alphabets are known as asymmetrical alphabets.

Q84.

In a book store, the words of the glowsign board "MODERN BOOK STORES" are individually flashed after 5/2, 17/4 and 41/8 seconds respectively. Each word is put off after a second. What is the least time after which full name of the book store can be read?

CAT 2002 · QA

Passage / Data

Answer the following question based on the information given below.

A boy is supposed to put a mango into a basket if ordered 1, an orange if ordered 2 and an apple if ordered 3. He took out 1 mango and 1 orange if ordered 4. He was given the following sequence of orders.

12332142314223314113234

Q85.

Each question is followed by two statements A and B. Answer each question using the following instructions:

Answer (1) if the question can be solved by any one of the statements, but not the other one.
Answer (2) if the question can be solved by using either of the two statements.
Answer (3) if the question can be solved by using both the statements together and not by any one of them.
Answer (4) if the question cannot be solved with the help of the given data and more data is required.

Four students were added to a dance class. Would the teacher be able to divide her students evenly into a dance team (or teams) of 8?

A. If 12 students were added, then the teacher could put everyone in teams of 8 without any left overs.
B. The number of students in the class is currently not divisible by 8.

CAT 2002 · QA

Passage / Data

Answer the following question based on the information given below.

A boy is supposed to put a mango into a basket if ordered 1, an orange if ordered 2 and an apple if ordered 3. He took out 1 mango and 1 orange if ordered 4. He was given the following sequence of orders.

12332142314223314113234

Q86.

Each question is followed by two statements A and B. Answer each question using the following instructions:

Answer (1) if the question can be solved by any one of the statements, but not the other one.
Answer (2) if the question can be solved by using either of the two statements.
Answer (3) if the question can be solved by using both the statements together and not by any one of them.
Answer (4) if the question cannot be solved with the help of the given data and more data is required.

Is x = y?

A. (x + y) $(\frac{1}{x} + \frac{1}{y})$ = 4

B. (x − 50)² = (y − 50)²

CAT 2002 · QA

Passage / Data

Answer the following question based on the information given below.

A boy is supposed to put a mango into a basket if ordered 1, an orange if ordered 2 and an apple if ordered 3. He took out 1 mango and 1 orange if ordered 4. He was given the following sequence of orders.

12332142314223314113234

Q87.

Each question is followed by two statements A and B. Answer each question using the following instructions:

Answer (1) if the question can be solved by any one of the statements, but not the other one.
Answer (2) if the question can be solved by using either of the two statements.
Answer (3) if the question can be solved by using both the statements together and not by any one of them.
Answer (4) if the question cannot be solved with the help of the given data and more data is required.

Is |x − 2| < 1?

A. |x| > 1
B. |x − 1| < 2

CAT 2001 · QA

Q88.

Let x, y and z be distinct integers. x and y are odd and positive, and z is even and positive. Which one of the following statements cannot be true?

CAT 2001 · QA

Q89.

A red light flashes 3 times per minute and a green light flashes 5 times in two minutes at regular intervals. If both lights start flashing at the same time, how many times do they flash together in each hour?

CAT 2001 · QA

Q90.

Of 128 boxes of oranges, each box contains at least 120 and at most 144 oranges. The number of boxes containing the same number of oranges is at least

CAT 2001 · QA

Q91.

In a 4-digit number, the sum of the first two digits is equal to that of the last two digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other two digits. What is the third digit of the number?

CAT 2001 · QA

Q92.

Anita had to do a multiplication. Instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?

CAT 2001 · QA

Passage / Data

Answer the following question based on the information given below.

The batting average (BA) of a test batsman is computed from runs scored and innings played-completed innings and incomplete innings (not out) in the following manner:

r₁ = number of runs scored in completed innings; n₁ = number of completed innings

r₂ = number of runs scored in incomplete innings; n₂ = number of incomplete innings

BA = $\frac{r_{1} + r_{2}}{n_{1}}$

To better assess batsman's accomplishments, the ICC is considering two other measures MBA₁ and MBA₂ defined as follows:

MBA₁ = $\frac{r_{1}}{n_{1}} + \frac{n_{2}}{n_{1}} \times m a x [0, (\frac{r_{2}}{n_{2}} - \frac{r_{1}}{n_{1}}])$

MBA₂ = $\frac{r_{1} + r_{2}}{n_{1} + n_{2}}$

Q93.

Let b be a positive integer and a = b² – b. If b ≥ 4, then a² – 2a is divisible by

CAT 2001 · QA

Passage / Data

Answer the following question based on the information given below.

The batting average (BA) of a test batsman is computed from runs scored and innings played-completed innings and incomplete innings (not out) in the following manner:

r₁ = number of runs scored in completed innings; n₁ = number of completed innings

r₂ = number of runs scored in incomplete innings; n₂ = number of incomplete innings

BA = $\frac{r_{1} + r_{2}}{n_{1}}$

To better assess batsman's accomplishments, the ICC is considering two other measures MBA₁ and MBA₂ defined as follows:

MBA₁ = $\frac{r_{1}}{n_{1}} + \frac{n_{2}}{n_{1}} \times m a x [0, (\frac{r_{2}}{n_{2}} - \frac{r_{1}}{n_{1}}])$

MBA₂ = $\frac{r_{1} + r_{2}}{n_{1} + n_{2}}$

Q94.

Ashish is given Rs. 158 in one rupee denominations. He has been asked to allocate them into a number of bags such that any amount required between Re. 1 and Rs. 158 can be given by handing out a certain number of bags without opening them. What is the minimum number of bags required?

CAT 2001 · QA

Passage / Data

Answer the following question based on the information given below.

The batting average (BA) of a test batsman is computed from runs scored and innings played-completed innings and incomplete innings (not out) in the following manner:

r₁ = number of runs scored in completed innings; n₁ = number of completed innings

r₂ = number of runs scored in incomplete innings; n₂ = number of incomplete innings

BA = $\frac{r_{1} + r_{2}}{n_{1}}$

To better assess batsman's accomplishments, the ICC is considering two other measures MBA₁ and MBA₂ defined as follows:

MBA₁ = $\frac{r_{1}}{n_{1}} + \frac{n_{2}}{n_{1}} \times m a x [0, (\frac{r_{2}}{n_{2}} - \frac{r_{1}}{n_{1}}])$

MBA₂ = $\frac{r_{1} + r_{2}}{n_{1} + n_{2}}$

Q95.

In some code, letters, a, b, c, d and e represent numbers 2, 4, 5, 6 and 10. However, we don't know which letter represent which number. Consider the following relationships:

i. a + c = e
ii. b – d = d
iii. e + a = b

CAT 2001 · QA

Passage / Data

Answer the following question based on the information given below.

The petrol consumption rate of a new model car 'Palto' depends on its speed and may be described by the graph below

Q96.

Choose 1; if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Choose 2; if the question can be answered by using either statement alone.
Choose 3; if the question can be answered by using both statements together, but cannot be answered using either statement alone.
Choose 4; if the question cannot be answered even by using both statements together.

What are the values of m and n?

n is an even integer, m is an odd integer, and m is greater than n.
Product of m and n is 30.

CAT 2001 · QA

Passage / Data

Answer the following question based on the information given below.

The petrol consumption rate of a new model car 'Palto' depends on its speed and may be described by the graph below

Q97.

Choose 1; if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Choose 2; if the question can be answered by using either statement alone.
Choose 3; if the question can be answered by using both statements together, but cannot be answered using either statement alone.
Choose 4; if the question cannot be answered even by using both statements together.

What is the value of X?

X and Y are unequal even integers, less than 10, and X/Y is an odd integer.
X and Y are even integers, each less than 10, and product of X and Y is 12.

CAT 2000 · QA

Q98.

Let D be a recurring decimal of the form, D = 0.a₁a₂a₁a₂a₁a₂ ......., where digits a₁ and a₂ lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?

CAT 2000 · QA

Q99.

Let S be the set of integers x such that

(i) 100 < x < 200

(ii) x is odd

(iii) x is divisible by 3 but not by 7

How many elements does S contain?

CAT 2000 · QA

Q100.

Let x, y and z be distinct integers, that are odd and positive. Which one of the following statements cannot be true?

CAT 2000 · QA

Q101.

Let S be the set of prime numbers greater than or equal to 2 and less than 100. Multiply all elements of S. With how many consecutive zeros will the product end?

CAT 2000 · QA

Q102.

Let N = 1421 × 1423 × 1425. What is the remainder when N is divided by 12?

CAT 2000 · QA

Q103.

The integers 34041 and 32506 when divided by a three-digit integer ‘n’ leave the same remainder. What is ‘n’?

CAT 2000 · QA

Q104.

Each of the numbers x₁, x₂...., x_n, n > 4, is equal to 1 or –1. Suppose,

x₁x₂x₃x₄ + x₂x₃x₄x₅ + x₃x₄x₅x₆ + ... + x_n–3x_n–2x_n–1x_n + x_n–2x_n–1x_nx₁+ x_n–1x_nx₁x₂ + x_nx₁x₂x₃= 0, then,

CAT 2000 · QA

Passage / Data

Answer the following question based on the information given below.

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.

The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

Q105.

Let N = 55³+ 17³ – 72³. N is divisible by

CAT 2000 · QA

Passage / Data

Answer the following question based on the information given below.

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.

The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

Q106.

Choose 1; if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Choose 2; if the question can be answered by using either statement alone.
Choose 3; if the question can be answered by using both statements together, but cannot be answered using either statement alone.
Choose 4; if the question cannot be answered even by using both statements together.

Let X be a real number. Is the modulus of X necessarily less than 3?

X(X + 3) < 0
X(X – 3) > 0

CAT 2000 · QA

Passage / Data

Answer the following question based on the information given below.

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.

The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

Q107.

Choose 1; if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone.
Choose 2; if the question can be answered by using either statement alone.
Choose 3; if the question can be answered by using both statements together, but cannot be answered using either statement alone.
Choose 4; if the question cannot be answered even by using both statements together.

What are the ages of two individuals, X and Y?

The age difference between them is 6 years.
The product of their ages is divisible by 6.

CAT 1999 · QA

Q108.

The number of positive integer valued pairs (x, y) satisfying 4x – 17y = 1 and x ≤ 1000 is

CAT 1999 · QA

Q109.

Let a, b, c be distinct digits. Consider a two-digit number ‘ab’ and a three-digit number ‘ccb’, both defined under the usual decimal number system, if (ab)² = ccb > 300, then the value of b is

CAT 1999 · QA

Q110.

The remainder when 7⁸⁴ is divided by 342 is

CAT 1999 · QA

Q111.

If n = 1 + x where x is the product of four consecutive positive integers, then which of the following
is/are true?

A. n is odd
B. n is prime
C. n is a perfect square

CAT 1999 · QA

Q112.

If n² = 12345678987654321, what is n?

CAT 1999 · QA

Passage / Data

Directions: Answer the questions based on the following information.
Recently, Ghosh Babu spent his winter vacation on Kyakya Island. During the vacation, he visited the local casino where he came across a new card game. Two players, using a normal deck of 52 playing cards, play this game. One player is called the ‘dealer’ and the other is called the ‘player’. First, the player picks a card at random from the deck. This is called the base card. The amount in rupees equal to the face value of the base card is called the base amount. The face values of ace, king, queen and jack are ten. For other cards the face value is the number on the card. Once the ‘player’ picks a card from the deck, the ‘dealer’ pays him the base amount. Then the ‘dealer’ picks a card from the deck and this card is called the top card. If the top card is of the same suit as the base card, the ‘player’ pays twice the base amount to the ‘dealer’. If the top card is of the same colour as the base card (but not the same suit), then the ‘player’ pays the base amount to the ‘dealer’. If the top card happens to be of a different colour than the base card, the ‘dealer’ pays the base amount to the ‘player’.
Ghosh Babu played the game four times. First time he picked eight of clubs and the ‘dealer’ picked queen of clubs. Second time, he picked ten of hearts and the ‘dealer’ picked two of spades. Next time, Ghosh Babu picked six of diamonds and the ‘dealer’ picked ace of hearts. Lastly, he picked eight of spades and the ‘dealer’ picked jack of spades. Answer the following questions based on these four games.

Q113.

Directions: Each question is followed by two statements I and II. Mark:
1. if the question can be answered by any one of the statements alone, but cannot be answered by using the other statement alone.
2. if the question can be answered by using either statement alone.
3. if the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
4. if the question cannot be answered even by using both the statements together.

Three professors A, B and C are separately given three sets of numbers to add. They were expected to find the answers to 1 + 1, 1 + 1 + 2, and 1 + 1 respectively. Their respective answers were 3, 3 and 2. How many of the professors are mathematicians?
I. A mathematician can never add two numbers correctly, but can always add three numbers correctly.
II. When a mathematician makes a mistake in a sum, the error is +1 or –1.

CAT 1998 · QA

Q114.

n³ is odd. Which of the following statement(s) is(are) true?

I. n is odd.
II. n² is odd.
III. n² is even.

CAT 1998 · QA

Passage / Data

âââââââDirection: Answer the questions based on the following information.

Production pattern for number of units (in cubic feet) per day.

For a truck that can carry 2,000 cubic ft, hiring cost per day is Rs. 1,000. Storing cost per cubic feet is Rs. 5 per day.

Q115.

(BE)² = MPB, where B, E, M and P are distinct integers. Then M =

CAT 1998 · QA

Passage / Data

âââââââDirection: Answer the questions based on the following information.

Production pattern for number of units (in cubic feet) per day.

For a truck that can carry 2,000 cubic ft, hiring cost per day is Rs. 1,000. Storing cost per cubic feet is Rs. 5 per day.

Q116.

Five-digit numbers are formed using only 0, 1, 2, 3, 4 exactly once. What is the difference between the maximum and minimum number that can be formed?

CAT 1998 · QA

Passage / Data

Answer the next 2 questions based on the following information.

A cow is tethered at point A by a rope. Neither the rope nor the cow is allowed to enter ΔABC.

∠BAC = 30°
I(AB) = I(AC) = 10 m

Q117.

Three wheels can complete 60, 36 and 24 revolutions per minute. There is a red spot on each wheel that touches the ground at time zero. After how much time, all these spots will simultaneously touch the ground again?

CAT 1998 · QA

Passage / Data

Answer the next 2 questions based on the following information.

A cow is tethered at point A by a rope. Neither the rope nor the cow is allowed to enter ΔABC.

∠BAC = 30°
I(AB) = I(AC) = 10 m

Q118.

A certain number, when divided by 899, leaves a remainder 63. Find the remainder when the same number is divided by 29.

CAT 1998 · QA

Passage / Data

Answer the next 2 questions based on the following information.

A cow is tethered at point A by a rope. Neither the rope nor the cow is allowed to enter ΔABC.

∠BAC = 30°
I(AB) = I(AC) = 10 m

Q119.

A is the set of positive integers such that when divided by 2, 3, 4, 5, 6 leaves the remainders 1, 2, 3, 4, 5 respectively. How many integers between 0 and 100 belong to set A?

CAT 1998 · QA

Passage / Data

Direction: Answer the question based on the following information.

A, B, C and D collected one-rupee coins following the given pattern.

Together they collected 100 coins.
Each one of them collected even number of coins.
Each one of them collected at least 10 coins.
No two of them collected the same number of coins.

Q120.

The maximum number of coins collected by any one of them cannot exceed

CAT 1998 · QA

Passage / Data

Direction: Answer the question based on the following information.

A, B, C and D collected one-rupee coins following the given pattern.

Together they collected 100 coins.
Each one of them collected even number of coins.
Each one of them collected at least 10 coins.
No two of them collected the same number of coins.

Q121.

If A collected 54 coins, then the difference in the number of coins between the one who collected maximum number of coins and the one who collected the second highest number of coins must be at least

CAT 1998 · QA

Passage / Data

Direction: Answer the question based on the following information.

A, B, C and D collected one-rupee coins following the given pattern.

Together they collected 100 coins.
Each one of them collected even number of coins.
Each one of them collected at least 10 coins.
No two of them collected the same number of coins.

Q122.

If A collected 54 coins and B collected two more coins than twice the number of coins collected by C, then the number of coins collected by B could be

CAT 1998 · QA

Passage / Data

Direction: Answer the question based on the following information.

A, B, C and D collected one-rupee coins following the given pattern.

Together they collected 100 coins.
Each one of them collected even number of coins.
Each one of them collected at least 10 coins.
No two of them collected the same number of coins.

Q123.

Number of students who have opted for subjects A, B and C are 60, 84 and 108 respectively. The examination is to be conducted for these students such that only the students of the same subject are allowed in one room. Also the number of students in each room must be same. What is the minimum number of rooms that should be arranged to meet all these conditions?

CAT 1998 · QA

Passage / Data

Direction: Answer the question based on the following information.

A, B, C and D collected one-rupee coins following the given pattern.

Together they collected 100 coins.
Each one of them collected even number of coins.
Each one of them collected at least 10 coins.
No two of them collected the same number of coins.

Q124.

You can collect as many rubies and emeralds as you can. Each ruby is worth Rs. 4 crore and each emerald is worth Rs. 5 crore. Each ruby weighs 0.3 kg. And each emerald weighs 0.4 kg. Your bag can carry at the most 12 kg. What should you collect to get the maximum wealth?

CAT 1998 · QA

Passage / Data

Answer the next 2 questions based on the following information.

A company purchases components A and B from Germany and USA respectively. A and B form 30% and 50% of the total production cost. Current gain is 20%. Due to change in the international scenario, cost of the German mark increased by 30% and that of USA dollar increased by 22%. Due to market conditions, the selling price cannot be increased beyond 10%.

Q125.

What is the digit in the unit’s place of 2⁵¹?

CAT 1998 · QA

Passage / Data

Answer the next 2 questions based on the following information.

A company purchases components A and B from Germany and USA respectively. A and B form 30% and 50% of the total production cost. Current gain is 20%. Due to change in the international scenario, cost of the German mark increased by 30% and that of USA dollar increased by 22%. Due to market conditions, the selling price cannot be increased beyond 10%.

Q126.

A number is formed by writing first 54 natural numbers in front of each other as 12345678910111213 ... Find the remainder when this number is divided by 8.

CAT 1998 · QA

Passage / Data

Answer the next 2 questions based on the following information.

A company purchases components A and B from Germany and USA respectively. A and B form 30% and 50% of the total production cost. Current gain is 20%. Due to change in the international scenario, cost of the German mark increased by 30% and that of USA dollar increased by 22%. Due to market conditions, the selling price cannot be increased beyond 10%.

Q127.

Direction: Each question is followed by two statements, I and II. Answer the questions based on the statements and mark the answer as
1. if the question can be answered with the help of any one statement alone but not by the other statement.
2. if the question can be answered with the help of either of the statements taken individually.
3. if the question can be answered with the help of both statements together.
4. if the question cannot be answered even with the help of both statements together.

Is n odd?
I. n is divisible by 3, 5, 7 and 9.
II. 0 < n < 400

CAT 1997 · QA

Passage / Data

Answer the next 3 questions based on the following information.

There are 60 students in a class. These students are divided into three groups A, B and C of 15, 20 and 25 students each. The groups A and C are combined to form group D.

Q128.

If n is an integer, how many values of n will give an integral value of $\frac{(16 n^{2} + 7 n + 6)}{n}$ ?

CAT 1997 · QA

Passage / Data

Answer the next 3 questions based on the following information.

There are 60 students in a class. These students are divided into three groups A, B and C of 15, 20 and 25 students each. The groups A and C are combined to form group D.

Q129.

P and Q are two positive integers such that PQ = 64. Which of the following cannot be the value of P + Q?

CAT 1997 · QA

Passage / Data

Answer the next 3 questions based on the following information.

There are 60 students in a class. These students are divided into three groups A, B and C of 15, 20 and 25 students each. The groups A and C are combined to form group D.

Q130.

If m and n are integers divisible by 5, which of the following is not necessarily true?

CAT 1997 · QA

Passage / Data

Direction: Answer the questions based on the following information.

A survey of 200 people in a community who watched at least one of the three channels — BBC, CNN and DD — showed that 80% of the people watched DD, 22% watched BBC and 15% watched CNN.

Q131.

P, Q and R are three consecutive odd numbers in ascending order. If the value of three times P is 3 less than two times R, find the value of R.

CAT 1997 · QA

Passage / Data

Direction: Answer the questions based on the following information.

For these questions the following functions have been defined.

la(x, y, z) = min(x + y, y + z)

le(x, y, z) = max (x − y, y − z)

ma(x, y, z) = $\frac{1}{2}$ [le(x, y, z) + la(x, y, z)]

Q132.

ABC is a three-digit number in which A > 0. The value of ABC is equal to the sum of the factorials of its three digits. What is the value of B?

CAT 1996 · QA

Passage / Data

Direction: Answer the questions based on the following information.

A, S, M and D are functions of x and y, and they are defined as follows.

A(x, y) = x + y

S(x, y) = x – y

M(x, y) = xy

D(x, y) = $\frac{x}{y}$ , y ≠ 0

Q133.

If n is any odd number greater than 1, then n(n² – 1) is

CAT 1996 · QA

Passage / Data

Answer the questions based on the following information.

A series S₁ of five positive integers is such that the third term is half the first term and the fifth term is 20 more than the first term. In series S₂, the n^th term defined as the difference between the (n + 1)^th term and the n^th term of series S₁, is an arithmetic progression with a common difference of 30.

Q134.

If a number 774958A96B is to be divisible by 8 and 9, the respective values of A and B will be

CAT 1995 · QA

Q135.

5⁶ - 1 is divisible by

CAT 1995 · QA

Passage / Data

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Q136.

72 hens cost Rs.__ 96.7__. Then what does each hen cost, where two digits in place of ‘__’ are not visible or are written in illegible hand?

CAT 1995 · QA

Passage / Data

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Q137.

The value of $\frac{55^{3} + 45^{3}}{55^{2} - 55 \times 45 + 45^{2}}$ is

CAT 1995 · QA

Passage / Data

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Q138.

For the product n(n + 1)(2n + 1), n ∈ N, which one of the following is not necessarily true?

CAT 1995 · QA

Passage / Data

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Q139.

The remainder obtained when a prime number greater than 6 is divided by 6 is

CAT 1995 · QA

Passage / Data

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Q140.

Three bells chime at an interval of 18 min, 24 min and 32 min. At a certain time they begin to chime together. What length of time will elapse before they chime together again?

CAT 1994 · QA

Passage / Data

Answer the next 3 questions based on the information given below:

Ghoshbabu is staying at Ghosh Housing Society, Aghosh Colony, Dighospur, Calcutta. In Ghosh Housing Society 6 persons read daily Ganashakti and 4 read Anand Bazar Patrika; in his colony there is no person who reads both. Total number of persons who read these two newspapers in Aghosh Colony and Dighospur is 52 and 200 respectively. Number of persons who read Ganashakti in Aghosh Colony and Dighospur is 33 and 121 respectively; while the persons who read Anand Bazar Patrika in Aghosh Colony and Dighospur are 32 and 117 respectively.

Q141.

If a + b + c = 0, where a ≠ b ≠ c, then $\frac{a^{2}}{2 a^{2} + b c} + \frac{b^{2}}{2 b^{2} + a c} + \frac{c^{2}}{2 c^{2} + a b}$ is equal to

CAT 1994 · QA

Passage / Data

Answer the next 2 questions based on the following information:

If
md(x) = x ,
mn(x,y) = minimum of x and y and
Ma(a,b,c,...) = maximum of a,b,c…

Q142.

It takes the pendulum of a clock 7 seconds to strike 4 o’clock. How much time will it take to strike 11 o’clock?

CAT 1994 · QA

Passage / Data

Answer the next 2 questions based on the following information:

If
md(x) = x ,
mn(x,y) = minimum of x and y and
Ma(a,b,c,...) = maximum of a,b,c…

Q143.

What is the smallest number which when increased by 5 is completely divisible by 8, 11 and 24?

CAT 1994 · QA

Passage / Data

Answer the next 3 questions based on the information given below:

Alphonso, on his death bed, keeps half his property for his wife and divide the rest equally among his three sons Ben, Carl and Dave. Some years later Ben dies leaving half his property to his widow and half to his brothers Carl and Dave together, shared equally. When Carl makes his will he keeps half his property for his widow and the rest he bequeaths to his younger brother Dave. When Dave dies some years later, he keeps half his property for his widow and the remaining for his mother. The mother now has Rs. 1,575,000.

Q144.

Which is the least number that must be subtracted from 1856, so that the remainder when divided by 7, 12, and 16 is 4.

CAT 1993 · QA

Q145.

Suppose one wishes to find distinct positive integers x, y such that (x + y)/xy is also a positive integer. Identify the correct alternative.

CAT 1993 · QA

Q146.

Given odd positive integers x, y and z, which of the following is not necessarily true?

CAT 1993 · QA

Passage / Data

Use the following information:

Eighty five children went to an amusement park where they could ride on the merry – go round, roller coaster, and Ferris wheel. It was known that 20 of them took all three rides, and 55 of them took at least two of the three rides. Each ride cost Re.1, and the total receipt of the amusement park was Rs.145.

Q147.

The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5 is

CAT 1993 · QA

Passage / Data

The following functions have been defined for three numbers A, B and C:

@ (A, B) = average of A and B
*(A, B) = product of A and B
/(A, B) = A divided by B

Answer these questions with the above data.

Q148.

Let x < 0, 0 < y < 1, z > 1. Which of the following may be false?

CAT 1993 · QA

Passage / Data

Directions for next 3 questions:

Q started to move from point B towards point A exactly an hour after P started from A in the opposite direction. Q’s speed was twice that of P. When P had covered one-sixth of the distance between the points A and B, Q had also covered the same distance.

Q149.

The smallest number which when divided by 4, 6 or 7 leaves a remainder of 2, is

CAT 1993 · QA

Passage / Data

Directions for next 3 questions:

Q started to move from point B towards point A exactly an hour after P started from A in the opposite direction. Q’s speed was twice that of P. When P had covered one-sixth of the distance between the points A and B, Q had also covered the same distance.

Q150.

The product of all integers from 1 to 100 will have the following numbers of zeros at the end.

CAT 1993 · QA

Passage / Data

Directions for next 3 questions:

Q started to move from point B towards point A exactly an hour after P started from A in the opposite direction. Q’s speed was twice that of P. When P had covered one-sixth of the distance between the points A and B, Q had also covered the same distance.

Q151.

Let x, y and z be distinct positive integers satisfying x < y < z and x + y + z = k. What is the smallest value of K that does not determine x, y, z uniquely?

CAT 1991 · QA

Passage / Data

Use the following information:

Prakash has to decide whether or not to test a batch of 1000 widgets before sending them to the buyer. In case he decides to test, he has two options: (a) Use test I ; (b) Use test II. Test I cost Rs. 2 per widget. However, the test is not perfect. It sends 20% of the bad ones to the buyer as good. Test II costs Rs. 3 per widget. It brings out all the bad ones. A defective widget identified before sending can be corrected at a cost of Rs. 25 per widget. All defective widgets are identified at the buyer’s end and penalty of Rs. 50 per defective widget has to be paid by Prakash.

Q152.

The number of integers n satisfying –n + 2 ≥ 0 and 2n ≥ 4 is

CAT 1991 · QA

Passage / Data

Use the following information:

Prakash has to decide whether or not to test a batch of 1000 widgets before sending them to the buyer. In case he decides to test, he has two options: (a) Use test I ; (b) Use test II. Test I cost Rs. 2 per widget. However, the test is not perfect. It sends 20% of the bad ones to the buyer as good. Test II costs Rs. 3 per widget. It brings out all the bad ones. A defective widget identified before sending can be corrected at a cost of Rs. 25 per widget. All defective widgets are identified at the buyer’s end and penalty of Rs. 50 per defective widget has to be paid by Prakash.

Q153.

The sum of two integers is 10 and the sum of their reciprocals is 5/12. Then the larger of these integers is

CAT 1991 · QA

Passage / Data

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.

Q154.

x, y and z are three positive integers such that x > y > z. Which of the following is closest to the product xyz?

CAT 1991 · QA

Passage / Data

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.

Q155.

What is the greatest power of 5 which can divide 80! exactly.

CAT 1991 · QA

Passage / Data

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.

Q156.

A third standard teacher gave a simple multiplication exercise to the kids. But one kid reversed the digits of both the numbers and carried out the multiplication and found that the product was exactly the same as the one expected by the teacher. Only one of the following pairs of numbers will fit in the description of the exercise. Which one is that?

CAT 1991 · QA

Passage / Data

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.

Q157.

Find the minimum integral value of n such that the division 55n/124 leaves no remainder.

CAT 1991 · QA

Passage / Data

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.

Q158.

Let k be a positive integer such that k + 4 is divisible by 7. Then the smallest positive integer n, greater than 2, such that k + 2n is divisible by 7 equals

CAT 1991 · QA

Passage / Data

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.

Q159.

In Sivakasi, each boy’s quota of match sticks to fill into boxes is not more than 200 per session. If he reduces the number of sticks per box by 25, he can fill 3 more boxes with the total number of sticks assigned to him. Which of the following is the possible number of sticks assigned to each boy?

CAT 1991 · QA

Passage / Data

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.

Q160.

If x is a positive integer such that 2x + 12 is perfectly divisible by x, then the number of possible values of x is

CAT 1991 · QA

Passage / Data

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.

Q161.

A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let p be a prime number greater than 5. Then (p² – 1) is

CAT 1991 · QA

Passage / Data

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as well as a library was one fourth the number of those having a laboratory alone. The number of schools having either a laboratory or a library or both was 35.

Q162.

To decide whether a n digit number is divisible by 7, we can define a process by which its magnitude is reduced as follows: (i₁, i₂, i₃, … , are the digits of the number, starting from the most significant digit).

i₁i₂ ……. i_n ⇒ i₁. 3^n-1 + i₂ . 3^n-2 + ……… + i_n . 3⁰.

e.g. 259 ⇒ 2.3² + 5.3¹ + 9.3⁰ = 18 + 15 + 9 = 42

Ultimately the resulting number will be seven after repeating the above process a certain number of times. After how many such stages, does the number 203 reduce to 7?

Number Theory — CAT Previous-Year Questions

Number Theory · CAT PYQs