If
15 × 14 × 13 × ··· × 3 × 2 × 1 = 3ᵐ × n
where m and n are positive integers, then what is the maximum value of m?
Answer & explanation
Answer: (b) 6
The largest power of 3 dividing 15! decides m. The multiples of 3 up to 15 give five 3s and 9 = 3² gives one more, so m can be at most 6.
- Count the factors of 3 in 15!: the multiples of 3 up to 15 are 3, 6, 9, 12, 15 — five numbers, one 3 each.
- 9 = 3² contributes one extra 3.
- Total power of 3 = ⌊15/3⌋ + ⌊15/9⌋ = 5 + 1 = 6, so the maximum m is 6 (n is then a whole number).
Remember · Highest power of prime p in n! = ⌊n/p⌋ + ⌊n/p²⌋ + ….
Question and answer: UPSC's official GS Paper II (2022, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·