HMMT 十一月 2013 · 冲刺赛 · 第 17 题

英文解析

17. [ 10 ] Find the sum of over all positive integers n with the property that the decimal representationnof terminates.1
    5 1 ni j
    Answer: The decimal representation of terminates if and only if n = 2 5 for some nonnegative
    2 nintegers i, j , so our desired sum is
    ∑ ∑ ∑ ∑
    2 5 5 − i − j − i − j − 1 − 1 − 1 − 1
    2 5 = 2 5 = (1 − 2 ) (1 − 5 ) = = .
    1 4 2
    i ≥ 0 j ≥ 0 i ≥ 0 j ≥ 0