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HMMT 十一月 2011 · 冲刺赛 · 第 1 题

HMMT November 2011 — Guts 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

[5] Determine the remainder when 1 + 2 + · · ·+ 2014 is divided by 2012.

解析

英文解析

  1. [ 5 ] Determine the remainder when 1 + 2 + · · · + 2014 is divided by 2012.
    Answer: 1009 We wish to find the value of 1 + 2 + · · · + 2014 modulo 2012. We have
    1 + 2 + · · · + 2014 = (2014)(2015) = 1007 · 2015 ≡ 1007 · 3 = 3021 ≡ 1009 (mod 2012) .1
    Remark: Note that, since 2 is not relatively prime to 2012, that this is not the same as 2
    (2)(3) ≡ 3 (mod 2012) .1 2