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HMMT 二月 2003 · 冲刺赛 · 第 18 题

HMMT February 2003 — Guts Round — Problem 18

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [8] Find the sum of the reciprocals of all the (positive) divisors of 144.
    HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND
    3 2

英文原题

[8] Find the sum of the reciprocals of all the (positive) divisors of 144.
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HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND

解析

英文解析

  1. Find the sum of the reciprocals of all the (positive) divisors of 144.
    Solution: 403 / 144
    As d ranges over the divisors of 144, so does 144 /d , so the sum of 1 /d is 1 / 144 times the sum of the divisors of 144. Using the formula for the sum of the divisors of a number
    (or just counting them out by hand), we get that this sum is 403, so the answer is
    403 / 144.
    3 24