HMMT 十一月 2008 · 冲刺赛 · 第 32 题

HMMT November 2008 — Guts Round — Problem 32

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