HMMT 十一月 2018 · 团队赛 · 第 3 题

HMMT November 2018 — Team Round — Problem 3

[ 25 ] For how many positive integers n ≤ 100 is it true that 10 n has exactly three times as many positive divisors as n has?

解析

[ 25 ] For how many positive integers n ≤ 100 is it true that 10 n has exactly three times as many positive divisors as n has? Proposed by: James Lin Answer: 28 a b Let n = 2 5 c , where 2 , 5 - c . Then, the ratio of the number of divisors of 10 n to the number of divisors a +2 b +2 1 − a of n is = 3. Solving for b , we find that b = . This forces ( a, b ) = (0 , 1) , (1 , 0). Therefore, a +1 b +1 2 a +1 the answers are of the form 2 k and 5 k whenever gcd( k, 10) = 1. There are 50 positive numbers of the form 2 k and 20 positive numbers of the form 5 k less than or equal to 100. Of those 70 numbers, only 1 4 1 4 · have k relatively prime to 10, so the answer is 70 · · = 28 . 2 5 2 5