HMMT 十一月 2012 · 冲刺赛 · 第 19 题

HMMT November 2012 — Guts Round — Problem 19

[ 11 ] Find the number of triples of nonnegative integers ( x, y, z ) such that 15 x + 21 y + 35 z = 525.

解析

[ 11 ] Answer: 21 First, note that 525 = 3 × 7 × 5 × 5. Then, taking the equation modulo 7 gives that ′ ′ ′ ′ 7 | x ; let x = 7 x for some nonnegative integer x . Similarly, we can write y = 5 y and z = 3 z for ′ ′ some nonnegative integers y , z . Then, after substitution and division of both sides by 105, the given ′ ′ ′ equation is equivalent to x + y + z = 5. This is the same as the problem of placing 2 dividers among ( ) 7 5 balls, so is = 21. 2