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HMMT 十一月 2011 · 冲刺赛 · 第 7 题

HMMT November 2011 — Guts Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 7 ] How many ordered triples of positive integers ( a, b, c ) are there for which a b c = 54000?
解析
  1. [ 7 ] How many ordered triples of positive integers ( a, b, c ) are there for which a b c = 54000? 4 3 3 a a a b b b 1 2 3 1 2 3 Answer: 16 We note that 54000 = 2 × 3 × 5 . Hence, we must have a = 2 3 5 , b = 2 3 5 , c c c 1 2 3 c = 2 3 5 . We look at each prime factor individually: • 4 a + 2 b + c = 4 gives 4 solutions: (1 , 0 , 0) , (0 , 2 , 0) , (0 , 1 , 2) , (0 , 0 , 4) 1 1 1 • 4 a + 2 b + c = 3 and 4 a + 2 b + c = 3 each give 2 solutions: (0 , 1 , 1) , (0 , 1 , 3). 2 2 2 3 3 3 Hence, we have a total of 4 × 2 × 2 = 16 solutions.