HMMT 二月 2003 · 冲刺赛 · 第 15 题

英文解析

15. The product of the digits of a 5-digit number is 180. How many such numbers exist?
    Solution: 360
    2 2
    Let the digits be a, b, c, d, e. Then abcde = 180 = 2 · 3 · 5 . We observe that thereare 6 ways to factor 180 into digits a, b, c, d, e (ignoring differences in ordering): 180 =
    1 · 1 · 4 · 5 · 9 = 1 · 1 · 5 · 6 · 6 = 1 · 2 · 2 · 5 · 9 = 1 · 2 · 3 · 5 · 6 = 1 · 3 · 3 · 4 · 5 = 2 · 2 · 3 · 3 · 5 .
    There are (respectively) 60, 30, 60, 120, 60, and 30 permutations of these breakdowns,
    for a total of 360 numbers. 1