HMMT 二月 2017 · 冲刺赛 · 第 5 题

英文解析

5. [ 6 ] Find the number of ordered triples of positive integers ( a, b, c ) such that
   6 a + 10 b + 15 c = 3000 .
   Proposed by: Yang Liu
   Answer: 4851
   Note that 6 a must be a multiple of 5, so a must be a multiple of 5. Similarly, b must be a multiple of
   3, and c must be a multiple of 2.
   ( )
   Set a = 5 A, b = 3 B, c = 2 C. Then the equation reduces to A + B + C = 100. This has = 485199
   solutions.2