HMMT 十一月 2015 · 冲刺赛 · 第 7 题

英文解析

7. [ 7 ] Let n be the smallest positive integer with exactly 2015 positive factors. What is the sum ofthe (not necessarily distinct) prime factors of n ? For example, the sum of the prime factors of 72 is
   2 + 2 + 2 + 3 + 3 = 14.
   Proposed by: Alexander Katz
   Answer: 116
   30 12 4
   Note that 2015 = 5 × 13 × 31 and that N = 2 · 3 · 5 has exactly 2015 positive factors. We claimthis is the smallest such integer. Note that N < 2 .66
   30 12 4 30 12 4
   If n has 3 distinct prime factors, it must be of the form p q r for some primes p, q, r , so n ≥ 2 · 3 · 5 .
   e f e + f
   If n has 2 distinct prime factors, it must be of the form p q > 2 where ( e + 1)( f + 1) = 2015. Itis easy to see that this means e + f > 66 so n > 2 > N .66
   2014
   If n has only 1 prime factor, we have n ≥ 2 > N .
   So N is the smallest such integer, and the sum of its prime factors is 2 · 30 + 3 · 12 + 5 · 4 = 116.