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HMMT 二月 2022 · 冲刺赛 · 第 5 题

HMMT February 2022 — Guts Round — Problem 5

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [6] Compute the remainder when 10002000400080016003200640128025605121024204840968192 is divided by 100020004000800160032.
解析
  1. [6] Compute the remainder when 10002000400080016003200640128025605121024204840968192 is divided by 100020004000800160032. Proposed by: Ankit Bisain Answer: 40968192 k Solution: Let X denote 2 except with leading zeroes added to make it four digits long. Let abc · · · k denote the number obtained upon concatenating a, b, c, . . . . We have 6 2 · X X . . . X = X X . . . X . 0 1 5 6 7 11 Therefore, X X . . . X divides X X . . . X , meaning the remainder when X X . . . X is divided by 0 1 5 0 1 11 0 1 13 X X . . . X is 0 1 5 X X = 40968192 . 12 13