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PUMaC 2020 · 数论（B 组） · 第 3 题

PUMaC 2020 — Number Theory (Division B) — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. Compute the last two digits of 9 + 9 + . . . + 9 .
解析
  1. Compute the last two digits of 9 + 9 + . . . + 9 . Proposed by: Nancy Xu Answer: 20 2 2020 2020 2020 2020 It is enough to compute the residue of 9 + 9 + . . . + 9 modulo 100. We have: 2020 2020 9 ≡ (10 − 1) (mod 100) ( ) 2020 ∑ 2020 n 2020 − n ≡ (10) ( − 1) (mod 100) n n =0 ( ) 2020 2019 2020 ≡ (10)( − 1) + ( − 1) (mod 100) 1 ≡ − 20200 + 1 (mod 100) ≡ 1 (mod 100) . k 2 2020 2020 k 2020 2020 2020 Then 9 ≡ 1 (mod 100) ≡ 1 (mod 100) for all k , so 9 + 9 + . . . + 9 ≡ 2020 ≡ 20 (mod 100). 1