Number System Question 4 - CAT 2023
Question
The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is:
Step-by-Step Explanation
- Interpret the factor count condition:
The problem states that the target numbers have exactly two distinct factors other than 1 and itself. Adding 1 and the number itself back into the count means we are looking for positive integers that have exactly 4 total factors.
- Identify the structural forms for exactly 4 factors:
A number N has exactly 4 factors if its prime factorization matches one of two cases:
- Case 1 (N = p³): Where p is a prime number.
Cubes of primes less than 50:
2³ = 8, 3³ = 27 ⇒ 2 numbers
- Case 2 (N = p₁ × p₂): Where p₁ and p₂ are distinct prime numbers.
Find all unique prime products less than 50:
- Products with 2: 2 × 3,5,7,11,13,17,19,23 ⇒ 8 numbers
- Products with 3: 3 × 5,7,11,13 ⇒ 4 numbers
- Products with 5: 5 × 7 ⇒ 1 number
Total for Case 2 = 8 + 4 + 1 = 13 numbers
Total overall positive integers = 2 + 13 = 15
Correct Answer: 15