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

Question

UPSC Quizroom · Accepted Answer

Step 1: Understand the problem

A number n has exactly two distinct factors other than 1 and n → total factors = 4 (1, n, and 2 others).

Numbers with exactly 4 positive divisors are either of the form:

p × q, where p and q are distinct primes → factors: 1, p, q, pq

p³, where p is prime → factors: 1, p, p², p³

Both cases give exactly 4 factors.

Step 2: List numbers < 50 of the form p × q (distinct primes)

Primes < 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Products < 50:

2 × 3 = 6

2 × 5 = 10

2 × 7 = 14

2 × 11 = 22

2 × 13 = 26

2 × 17 = 34

2 × 19 = 38

2 × 23 = 46

3 × 5 = 15

3 × 7 = 21

3 × 11 = 33

3 × 13 = 39

5 × 7 = 35

Other products exceed 50

Total from p × q = 13 numbers

Step 3: List numbers < 50 of the form p³ (prime cubed)

2³ = 8

3³ = 27

5³ = 125 > 50 → ignore

Total from p³ = 2 numbers

Step 4: Total numbers

Total = 13 + 2 = 15

✅ Answer

15

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