Number Theory — CAT Previous-Year Questions

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

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

Let x, y, and z be real numbers satisfying

4(x² + y² + z²) = a,

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

Then a equals

When 10¹⁰⁰ is divided by 7, the remainder is

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

When 3³³³ is divided by 11, the remainder is

If 10⁶⁸ is divided by 13, the remainder is

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

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

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

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

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

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?

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?

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

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

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

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

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:

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

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

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

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

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 

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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 

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

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?

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?

If x + 1 = x² and x > 0, then 2x⁴ is:

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

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

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?

What are the last two digits of 7²⁰⁰⁸?

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?

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?

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.

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

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?

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:

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

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

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

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?

The rightmost non-zero digit of the number 30²⁷²⁰ is

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

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?

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

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

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

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?

1. b is even
2. b is greater than 16

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

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

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

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

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

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

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, 
 

What is the remainder when 4⁹⁶ is divided by 6?

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?

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

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

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?

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?

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

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

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

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

The remainder when 2²⁵⁶ is divided by 17 is

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?

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? 

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.

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) $\left(\frac{1}{x}+\frac{1}{y}\right)$ = 4

B. (x − 50)² = (y − 50)²

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

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?

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?

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

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?

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?

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

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?

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

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?

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

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?

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

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?

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?

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

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?

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

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

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,

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

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?

1. X(X + 3) < 0
2. X(X – 3) > 0

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?

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

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

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

The remainder when 7⁸⁴ is divided by 342 is

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

If n² = 12345678987654321, what is n?

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.

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

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

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

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?

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?

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

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?

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

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

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

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?

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?

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

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.

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

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

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

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

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.

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?

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

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

5⁶ - 1 is divisible by

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?

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

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

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

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?

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

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?

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

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

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

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

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

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

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

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

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?

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

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

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

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

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?

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

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

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?

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

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

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).

i₁ i₂ ……. 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?