468
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