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HMMT 二月 2025 · 冲刺赛 · 第 7 题

HMMT February 2025 — Guts Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [6] The number 9 8 9 − 8 1001 is an integer. Compute the sum of its prime factors.
解析
  1. [6] The number 9 8 9 − 8 1001 is an integer. Compute the sum of its prime factors. Proposed by: Derek Liu Answer: 231 Solution: Observe 9 8 6 6 9 − 8 = 27 − 16 2 2 4 2 2 4 = (27 − 16 )(27 + 27 · 16 + 16 ) 2 2 2 2 = (27 − 16)(27 + 16)(27 − 27 · 16 + 16 )(27 + 27 · 16 + 16 ) = 11 · 43 · 553 · 1417 . The remaining factorizations are motivated by the fact that 1001 = 7 · 11 · 13. We see that 553 = 7 · 79 and 1417 = 13 · 109, so the answer is 43 + 79 + 109 = 231 .