Question

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

• A. 14
• B. 15
• C. 16
• D. 20

Answer 1

The Correct Option is B

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:

\(N = p_1 \times p_2\) (where \(p_1\) and \(p_2\) are prime numbers)

As we can see from Case I, the numbers that are perfect cubes of primes and less than 50 are 8 and 27 (2 numbers).

Numbers in the format \((2 \times 3), (2 \times 5), (2 \times 7), (2 \times 11), (2 \times 13), (2 \times 17), (2 \times 19), (2 \times 23), (3 \times 5), (3 \times 7), (3 \times 11), (3 \times 13), (5 \times 7)\) result from Case II (13 numbers).

Thus, the total number of numbers with two different factors is \((13 + 2) = 15\).