Question

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

Answer 1

Positive integers less than 50 that are either perfect cubes below 50 or integers that are products of exactly two separate primes are those that have exactly two distinct components other than 1 and themselves.
Case I: \(N = p ^3\) (where p is a prime number)
Case II:\( N = p_ 1​× p _2​\)(where \( p_ 1​, p_ 2\) ​are the prime numbers)
As we can see from Case I, 8 and 27 are the numbers that are a cube of prime and less than 50 (2 numbers).
Numbers in the format \((2\times3), (2\times5), (2\times7), (2\times11), (2\times13), (2\times17), (2\times19), (2\times23), (3\times5), (3\times7), (3\times11), (3\times13), (5\times7)\) {(13 numbers)} will be obtained from Case II.
Thus, \((13+2) = 15\) is the total number of numbers with two different factors. 
So the correct answer is 15.