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HMMT 二月 2005 · 代数 · 第 9 题

HMMT February 2005 — Algebra — Problem 9

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. The number 27,000,001 has exactly four prime factors. Find their sum. 4 3 2
解析
  1. The number 27,000,001 has exactly four prime factors. Find their sum. Solution: 652 3 First, we factor 6 2 3 27 x + 1 = (3 x ) + 1 2 4 2 = (3 x + 1)(9 x − 3 x + 1) 2 4 2 2 = (3 x + 1)((9 x + 6 x + 1) − 9 x ) 2 2 2 2 = (3 x + 1)((3 x + 1) − (3 x ) ) 2 2 2 = (3 x + 1)(3 x + 3 x + 1)(3 x − 3 x + 1) . Letting x = 10, we get that 27000001 = 301 · 331 · 271. A quick check shows that 301 = 7 · 43, so that 27000001 = 7 · 43 · 271 · 331. Each factor here is prime, and their sum is 652. 4 3 2