Number Theory — XAT Previous-Year Questions

The least common multiple of a number and 990 is 6930. The greatest common divisor of that number and 550 is 110. What is the sum of the digits of the least possible value of that number?

Consider a 4-digit number of the form abbb, i.e., the first digit is a (a > 0) and the last three digits are all b. Which of the following conditions is both NECESSARY and SUFFICIENT to ensure that the 4-digit number is divisible by a?

Let x and y be two positive integers and p be a prime number. If x(x – p) – y(y + p) = 7p, what will be the minimum value of x – y? 

A supplier receives orders from 5 different buyers. Each buyer places their order only on a Monday. The first buyer places the order after every 2 weeks, the second buyer, after every 6 weeks, the third buyer, after every 8 weeks, the fourth buyer, every 4 weeks, and the fifth buyer, after every 3 weeks. It is known that on January 1st, which was a Monday, each of these five buyers placed an order with the supplier.

On how many occasions, in the same year, will these buyers place their orders together excluding the order placed on January 1st?

The sum of the cubes of two numbers is 128, while the sum of the reciprocals oftheir cubes is 2. What is the product of the squares of the numbers?

Nadeem’s age is a two-digit number X, squaring which yields a three-digit number,whose last digit is Y. Consider the statements below:

Statement I: Y is a prime number
Statement II: Y is one-third of X

To determine Nadeem’s age uniquely:

Wilma, Xavier, Yaska and Zakir are four young friends, who have a passion for integers. One day, each of them selects one integer and writes it on a wall. The writing on the wall shows that Xavier and Zakir picked positive integers, Yaska picked a negative one, while Wilma’s integer is either negative, zero or positive. If their integers are denoted by the first letters of their respective names, the following is true:

W⁴ + X³ + Y² + Z ≤ 4

X³ + Z ≥ 2

W⁴ + Y² ≤ 2

Y² + Z ≥ 3

Given the above, which of these can W² + X² + Y² + Z² possibly evaluate to?

Fatima found that the profit earned by the Bala dosa stall today is a three-digit number. She also noticed that the middle digit is half of the leftmost digit, while the rightmost digit is three times the middle digit. She then randomly interchanged the digits and obtained a different number. This number was more than the original number by 198.

What was the middle digit of the profit amount?

The Madhura Fruits Company is packing four types of fruits into boxes. There are 126 oranges, 162 apples, 198 guavas and 306 pears. The fruits must be packed in such a way that a given box must have only one type of fruit and must contain thesame number of fruit units as any other box.

What is the minimum number of boxes that must be used?

What is the remainder if 19²⁰ – 20¹⁹ is divided by 7?

When expressed in a decimal form, which of the following numbers will be nonterminating as well as non-repeating?

Consider the four variables A, B, C and D and a function Z of these variables, Z = 15A² − 3B⁴ + C + 0.5D. It is given that A, B, C and D must be non-negative integers and that all of the following relationships must hold:
i) 2A + B ≤ 2
ii) 4A + 2B + C ≤ 12
iii) 3A + 4B + D ≤ 15 

If Z needs to be maximised, then what value must D take?

When opening his fruit shop for the day a shopkeeper found that his stock of apples could be perfectly arranged in a complete triangular array: that is, every row with one apple more than the row immediately above, going all the way up ending with a single apple at the top.

During any sales transaction, apples are always picked from the uppermost row, and going below only when that row is exhausted. When one customer walked in the middle of the day she found an incomplete array in display having 126 apples totally. How many rows of apples (complete and incomplete) were seen by this customer? (Assume that the initial stock did not exceed 150 apples.)

If x² + x + 1 = 0, then x²⁰¹⁸ + x²⁰¹⁹ equals which of the following: 

We have two unknown positive integers m and n, whose product is less than 100.
There are two additional statement of facts available:

1. mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }
2. m + n is a perfect square.

Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?

Find the value of the expression: 10 + 10³ + 10⁶ + 10⁹

X and Y are the digits at the unit's place of the numbers (408X) and (789Y) where X ≠ Y. However, the digits at the unit's place of the numbers (408X)⁶³ and (789Y)⁸⁵ are the same. What will be the possible value(s) of (X + Y)?

Example: If M = 3 then the digit at unit's place of the number (2M) is 3 (as the number is 23) and the digit at unit's place of the number (2M)² is 9 (as 23² is 529).

David has an interesting habit of spending money. He spends exactly £X on the X^th day of a month. For example, he spends exactly £5 on the 5^th of any month. On a few days in a year, David noticed that his cumulative spending during the last 'four consecutive days' can be expressed as 2^N where N is a natural number. What can be the possible value(s) of N?

An institute has 5 departments and each department has 50 students. If students are picked up randomly from all 5 departments to form a committee, what should be the minimum number of students in the committee so that at least one department should have representation of minimum 5 students?

If N = (11^p+7)(7^q-2)(5^r+1)(3^s) is a perfect cube, where p, q, r and s are positive integers, then the smallest value of p + q + r + s is:

For two positive integers a and b, if (a + b)^(a+b) is divisible by 500, then the least possible value of a × b is:

If a, b and c are 3 consecutive integers between –10 to +10 (both inclusive), how many integer values are possible for the expression $\frac{a^3+b^3+c^3+3abc}{{(a+b+c)}^2}$?

An ascending series of numbers satisfies the following conditions:

1. When divided by 3, 4, 5 or 6, the numbers leave a remainder of 2.
2. When divided by 11, the numbers leave no remainder.

The 6th number in this series will be:

If the last 6 digits of [(M)! – (N)!] are 999000, which of the following option is not possible for (M) × (M – N)?

Both (M) and (N) are positive integers and M > N. (M)! is factorial M.

A three-digit number has digits in strictly descending order and divisible by 10. By changing the places of the digits a new three-digit number is constructed in such a way that the new number is divisible by 10. The difference between the original number and the new number is divisible by 40. How many numbers will satisfy all these conditions?

Three Vice Presidents (VP) regularly visit the plant on different days. Due to labour unrest, VP (HR) regularly visits the plant after a gap of 2 days. VP (Operations) regularly visits the plant after a gap of 3 days. VP (Sales) regularly visits the plant after a gap of 5 days. The VPs do not deviate from their individual schedules. CEO of the company meets the VPs when all the three VPs come to the plant together. CEO is one leave from January 5th to January 28th, 2012. Last time CEO met the VPs on January 3, 2012. When is the next time CEO will meet all the VPs?