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PUMaC 2011 · 数论（A 组） · 第 3 题

PUMaC 2011 — Number Theory (Division A) — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. [ 4 ] What is the sum of all primes p such that 7 − 6 + 2 is divisible by 43?
解析
  1. Note that 7 = 343 ≡ − 1 (mod 43) and that 6 = (6 ) ≡ 1 (mod 43). Therefore, for p p p p p ≡ 0 , 1 , 2 , 3 , 4 , 5 (mod 6), 7 − 6 + 2 ≡ 2 , 3 , 15 , 0 , 32 , 3 (mod 43). Therefore, if 43 | 7 − 6 + 2, p ≡ 3 (mod 6). This means that p = 3 is the only solution, so that the sum of all solutions is 3.