Prime and Composite Numbers — CAT Previous-Year Questions

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

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?

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,

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, 
 

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

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