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HMMT 十一月 2011 · 团队赛 · 第 1 题

HMMT November 2011 — Team Round — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 2 ] Find the number of positive integers x less than 100 for whichx x x x x x x
    3 + 5 + 7 + 11 + 13 + 17 + 19
    is prime.
    3 2
解析

英文解析

  1. [ 2 ] Find the number of positive integers x less than 100 for whichx x x x x x x
    3 + 5 + 7 + 11 + 13 + 17 + 19
    is prime.
    Answer: 0 We claim that our integer is divisible by 3 for all positive integers x . Indeed, we havex x x x x x x x x x x x x x
    3 + 5 + 7 + 11 + 13 + 17 + 19 ≡ (0) + ( − 1) + (1) + ( − 1) + (1) + ( − 1) + (1)
    x x
    ≡ 3[(1) + ( − 1) ]
    ≡ 0 (mod 3) .
    It is clear that for all x ≥ 1, our integer is strictly greater than 3, so it will always be composite,
    making our answer 0.
    3 2