HMMT 十一月 2008 · 冲刺赛 · 第 32 题

HMMT November 2008 — Guts Round — Problem 32

个人笔记

[ 15 ] Pirate ships Somy and Lia are having a tough time. At the end of the year, they are both onepillage short of the minimum required for maintaining membership in the Pirate Guild, so they decideto pillage each other to bring their counts up. Somy by tradition only pillages 28 · 3 coins for integerskjk , and Lia by tradition only pillages 82 · 3 coins for integers j . Note that each pillage can have adifferent k or j . Soma and Lia work out a system where Somy pillages Lia n times, Lia pillages Somyn times, and after both sets of pillages Somy and Lia are financially even.
What is the smallest n can be? 2

解析

[ 15 ] Pirate ships Somy and Lia are having a tough time. At the end of the year, they are both onepillage short of the minimum required for maintaining membership in the Pirate Guild, so they decideto pillage each other to bring their counts up. Somy by tradition only pillages 28 · 3 coins for integerskjk , and Lia by tradition only pillages 82 · 3 coins for integers j . Note that each pillage can have adifferent k or j . Soma and Lia work out a system where Somy pillages Lia n times, Lia pillages Somyn times, and after both sets of pillages Somy and Lia are financially even.
What is the smallest n can be?
Answer: 2 Clearly, n = 1 cannot be acheived, because 28 · 3 is never a multiple of 82. However,ktwo pillages is enough: Somy pillages 28 and 28 · 81 from Lia, and Lia pillages 81 and 81 · 27 from
Somy. As is easily checked, both pillage 28 · 82. 2