HMMT 十一月 2011 · 冲刺赛 · 第 7 题

英文解析

7. [ 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.