CAT Number Systems Previous Year Questions (PYQs)

Suppose $a,b,c$ are three distinct natural numbers, such that $3ac = 8(a+b)$. Then, the smallest possible value of $3a+2b+c$ is

If $m$ and $n$ are integers such that $(m+2n)(2m+n) = 27$, then the maximum possible value of $2m-3n$ is

The sum of digits of the number $(625)^{65} \times (128)^{36}$, is

The number of divisors of $(2^6 \times 3^5 \times 5^3 \times 7^2)$, which are of the form $(3r+1)$, where $r$ is a non-negative integer, is

The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x > y \geq 3$ and $x + y < 14$ is

In a 3-digit number N, the digits are non-zero and distinct such that none of the digits is a perfect square, and only one of the digits is a prime number. Then, the number of factors of the minimum possible value of N is

When $10^{100}$ is divided by 7, the remainder is

For any natural number $n$, let $a_{n}$ be the largest integer not exceeding $\sqrt{n}$. Then the value of $a_{1}+a_{2}+\ldots .+a_{50}$ is

When $3^{333}$ is divide by 11, the remainder is

If $m$ and $n$ are natural numbers such that $n > 1$, and $m^{n}=2^{25} \times 3^{40}$, then $m-n$ equals

If $10^{68}$ is divided by $13$, the remainder is

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

Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km till then, where x is a whole number and is palindromic, i.e., $x$ remains unchanged when its digits are reversed. At the end of the trip, the car had traveled a total of $26862 \mathrm{~km}$ till then, this number again being palindromic. If Brishti never drove at more than $110 \mathrm{~km} / \mathrm{hr}$, then the greatest possible average speed at which she drove during the trip, in km/hr, was

For a 4-digit number (greater than 1000), sum of the digits in the thousands, hundreds, and tens places is 15. Sum of the digits in the hundreds, tens, and units places is 16. Also, the digit in the tens place is 6 more than the digit in the units place. The difference between the largest and smallest possible value of the number is

The sum of all the digits of the number $(10^{50}+10^{25}-123)$, is

For any natural numbers $m, n$, and $k$, such that $k$ divides both $m+2 n$ and $3 m$ $+4 n, k$ 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

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

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 numbers of the form $4^k + 3(4^k) + 4^{k + 2}$, where k is any positive integer. Then (A + B) equals

For some natural number $n$, assume that $(15000)!$ 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

How many three-digit numbers are greater than $100$ and increase by $198$ when the three digits are arranged in the reverse order?

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

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