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

Since there are two distinct factors other than 1, and itself, which implies the total number of factors of N is 4.

It can be done in two ways. 

First case: $$N\ =\ p^3$$ (where p is a prime number)

Second case: $$N\ =\ p_1\times\ p_2$$ (Where $$\ p_1,\ p_2$$ are the prime numbers)

From case 1, we can see that the numbers which is a cube of prime and less than 50 are 8, and 27 (2 numbers).

From case 2, we will get the numbers in the form (2*3), (2*5), (2*7), (2*11), (2*13), (2*17), (2*19), (2*23), (3*5), (3*7), (3*11), (3*13), (5*7) {(13 numbers)}

Hence, the total number of numbers having two distinct factors is (13+2) = 15.