Factors — CAT Previous-Year Questions

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

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

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

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?

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

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?

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?

5⁶ - 1 is divisible by

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