The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is:

Question

UPSC Quizroom · Accepted Answer

Step 1: Number of factors formula

If n = p₁^a₁ × p₂^a₂ × … × p_k^a_k, then the number of positive factors:

(a₁ + 1)(a₂ + 1)…(a_k + 1) = 15

Step 2: Factorize 15

15 = 15 × 1
15 = 5 × 3
15 = 3 × 5

So possible forms of n:

p^14 → (14 + 1) = 15

p^4 q^2 → (4 + 1)(2 + 1) = 15

p^2 q^4 → (2 + 1)(4 + 1) = 15

Step 3: Find smallest numbers

p^14 → 2^14 = 16384 (too large)

p^4 q^2 → smallest primes p=2, q=3 → 2^4 × 3^2 = 16 × 9 = 144

p^2 q^4 → p=2, q=3 → 2^2 × 3^4 = 4 × 81 = 324

So first number = 144, second number = 324

Step 4: Sum

144 + 324 = 468

Answer: 468

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