Question

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

Answer 1

Objective: Determine the count of positive integers below 50 that satisfy either of the following conditions:

• Condition 1: The integer is the perfect cube of a prime number, expressed as \(N = p^3\).
• Condition 2: The integer is the product of two distinct prime numbers, expressed as \(N = p_1 \times p_2\).

Condition 1 Analysis:
Identify primes whose cubes are less than 50:
- \(2^3 = 8\)
- \(3^3 = 27\)
Result for Condition 1: 2 numbers

Condition 2 Analysis:
Identify products of two distinct primes that are less than 50:
[ \(2 \times 3 = 6,\quad 2 \times 5 = 10,\quad 2 \times 7 = 14,\quad 2 \times 11 = 22,\quad 2 \times 13 = 26,\quad 2 \times 17 = 34,\quad 2 \times 19 = 38,\quad 2 \times 23 = 46,\) ]
[ \(3 \times 5 = 15,\quad 3 \times 7 = 21,\quad 3 \times 11 = 33,\quad 3 \times 13 = 39,\quad 5 \times 7 = 35\) ]
Result for Condition 2: 13 numbers

Total Count = 2 + 13 = 15

Final Answer: 15