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HMMT 十一月 2015 · 冲刺赛 · 第 8 题

HMMT November 2015 — Guts Round — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 7 ] For how many pairs of nonzero integers ( c, d ) with − 2015 ≤ c, d ≤ 2015 do the equations cx = dand dx = c both have an integer solution?
解析

英文解析

  1. [ 7 ] For how many pairs of nonzero integers ( c, d ) with − 2015 ≤ c, d ≤ 2015 do the equations cx = dand dx = c both have an integer solution?
    Proposed by: Yang Liu
    Answer: 8060
    We need both c/d and d/c to be integers, which is equivalent to | c | = | d | , or d = ± c . So there are 4030
    ways to pick c and 2 ways to pick d , for a total of 8060 pairs.