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HMMT 十一月 2018 · GEN 赛 · 第 1 题

HMMT November 2018 — GEN Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. What is the largest factor of 130000 that does not contain the digit 0 or 5?
解析
  1. What is the largest factor of 130000 that does not contain the digit 0 or 5? Proposed by: Farrell Eldrian Wu Answer: 26 If the number is a multiple of 5, then its units digit will be either 0 or 5. Hence, the largest such number must have no factors of 5. 4 4 We have 130000 = 2 · 5 · 13. Removing every factor of 5, we get that our number must be a factor of 4 2 · 13 = 208. If our number contains a factor of 13, we cancel the factor of 2 from 208, 104, and 52 until we get 26. 4 Otherwise, the largest number we can have is 2 = 16. We conclude that the answer is 26.