返回题库

HMMT 二月 2005 · 代数 · 第 9 题

HMMT February 2005 — Algebra — Problem 9

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. The number 27,000,001 has exactly four prime factors. Find their sum.
    4 3 2
解析

英文解析

  1. The number 27,000,001 has exactly four prime factors. Find their sum.
    Solution: 652
    First, we factor 3
    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 theirsum is 652.
    4 3 2