HMMT 十一月 2012 · 冲刺赛 · 第 11 题

英文解析

11. [ 8 ]
    Answer: 45 The number of zeroes that n ! ends with is the largest power of 10 dividing n !. Theexponent of 5 dividing n ! exceeds the exponent of 2 dividing n !, so we simply seek the exponent of 5
    dividing n !.
    For a number less than 125, this exponent is just the number of multiples of 5, but not 25, less than nplus twice the number of multiples of 25 less than n .
    Counting up, we see that 24! ends with 4 zeroes while 25! ends with 6 zeroes, so n ! cannot end with 5
    zeroes. Continuing to count up, we see that the smallest n such that n ! ends with 10 zeroes is 45.