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HMMT 十一月 2024 · 冲刺赛 · 第 4 题

HMMT November 2024 — Guts Round — Problem 4

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [6] The number 17 when written out in base 10 contains 8 distinct digits from 1 , 2 , . . . , 9 , with no repeated digits or zeroes. Compute the missing nonzero digit.
解析
  1. [6] The number 17 when written out in base 10 contains 8 distinct digits from 1 , 2 , . . . , 9, with no repeated digits or zeroes. Compute the missing nonzero digit. Proposed by: Karthik Venkata Vedula Answer: 8 Solution: Observe that 6 6 17 ≡ ( − 1) = 1 mod 9 . 6 If x is the missing digit, then the digits of 17 sum to (1 + 2 + · · · + 9) − x = 45 − x , so 45 − x ≡ 1 mod 9. We conclude x = 8 .