HMMT 二月 2026 · 冲刺赛 · 第 6 题

HMMT February 2026 — Guts Round — Problem 6

[6] Compute the largest positive integer n such that n ! divides ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 4 8 16 32 64 128 256 128 64 32 16 8 4 2 · · · · · · · . 128 64 32 16 8 4 2 1

解析

[6] Compute the largest positive integer n such that n ! divides ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 4 8 16 32 64 128 256 128 64 32 16 8 4 2 · · · · · · · . 128 64 32 16 8 4 2 1 Proposed by: Andrew Brahms, Carlos Rodriguez Answer: 256 Solution: We can expand out the product into the following: 2 4 8 16 32 64 128 256! 128! 64! 32! 16! 8! 4! 2! · · · · · · · 2 4 8 16 32 64 128 256 128! 64! 32! 16! 8! 4! 2! 1! After canceling, we see that the expression is equal to 256! . Thus, the answer is 256 .