Top 137 CAT Number Systems Questions With Solutions

Let a, b, m and n be natural numbers such that $$a>1$$ and $$b>1$$. If $$a^{m}b^{n}=144^{145}$$, then the largest possible value of $$n-m$$ 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

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

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

The number of coins collected per week by two coin-collectors A and B are in the ratio 3 : 4. If the total number of coins collected by A in 5 weeks is a multiple of 7, and the total number of coins collected by B in 3 weeks is a multiple of 24, then the minimum possible number of coins collected by A in one week is

The number of all natural numbers up to 1000 with non-repeating digits is

Consider six distinct natural numbers such that the average of the two smallest numbers is 14, and the average of the two largest numbers is 28. Then, the maximum possible value of the average of these six numbers 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 (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 all possible integers n satisfying $$2.25\leq2+2^{n+2}\leq202$$, then the number of integer values of $$3+3^{n+1}$$ 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

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

How many 4-digit numbers, each greater than 1000 and each having all four digits distinct, are there with 7 coming before 3?

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

How many integers in the set {100, 101, 102, ..., 999} have at least one digit repeated?

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?

How many pairs(a, b) of positive integers are there such that $$a\leq b$$ and $$ab=4^{2017}$$ ?