Question

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

Answer 1

Objective: Find the number of positive integers less than 50 that are either: 

• Case I: Perfect cubes of prime numbers: \(N = p^3\)
• Case II: Products of exactly two distinct prime numbers: \(N = p_1 \times p_2\)

Case I:
Primes whose cubes are less than 50:
- \(2^3 = 8\)
- \(3^3 = 27\)
⇒ 2 numbers

Case II:
Products of two distinct primes less than 50:
\[ 2 \times 3 = 6,\quad 2 \times 5 = 10,\quad 2 \times 7 = 14,\quad 2 \times 11 = 22,\quad 2 \times 13 = 26,\quad 2 \times 17 = 34,\quad 2 \times 19 = 38,\quad 2 \times 23 = 46, \] \[ 3 \times 5 = 15,\quad 3 \times 7 = 21,\quad 3 \times 11 = 33,\quad 3 \times 13 = 39,\quad 5 \times 7 = 35 \] ⇒ 13 numbers

Total = 2 + 13 = 15

Final Answer: 15