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HMMT 十一月 2016 · 冲刺赛 · 第 8 题

HMMT November 2016 — 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 ] Danielle picks a positive integer 1 ≤ n ≤ 2016 uniformly at random. What is the probability thatgcd( n, 2015) = 1?
解析

英文解析

  1. [ 7 ] Danielle picks a positive integer 1 ≤ n ≤ 2016 uniformly at random. What is the probability thatgcd( n, 2015) = 1?
    Proposed by: Evan Chen
    1441
    Answer:
    2016
    We split the interval [1 , 2016] into [1 , 2015] and 2016. The number of integers in [1 , 2015] that are
    4 12 30
    relatively prime to 2015 is φ (2015) = · · · 2015 = 1440. Also, 2016 is relatively prime to 2015, so
    5 13 31
    there are a total of 1441 numbers in [1 , 2016] that are relatively prime to 2015. Then the probability
    1441
    of picking a number relatively prime to 2015 is .
    2016