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HMMT 二月 2007 · 冲刺赛 · 第 7 题

HMMT February 2007 — Guts Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 7 ] A student at Harvard named Kevin Was counting his stones by 11 He messed up n times And instead counted 9s And wound up at 2007. How many values of n could make this limerick true?
解析
  1. [ 7 ] A student at Harvard named Kevin Was counting his stones by 11 He messed up n times And instead counted 9s And wound up at 2007. How many values of n could make this limerick true? Answer: 21 . The mathematical content is that 9 n + 11 k = 2007, for some nonnegative integers n and k . As 2007 = 9 · 223, k must be divisible by 9. Using modulo 11, we see that n is 3 more than a multiple of 11. Thus, the possibilities are n = 223 , 212 , 201 , . . . , 3, which are 21 in number.