HMMT 十一月 2015 · 冲刺赛 · 第 7 题

HMMT November 2015 — Guts Round — Problem 7

[ 7 ] Let n be the smallest positive integer with exactly 2015 positive factors. What is the sum of the (not necessarily distinct) prime factors of n ? For example, the sum of the prime factors of 72 is 2 + 2 + 2 + 3 + 3 = 14.

解析

[ 7 ] Let n be the smallest positive integer with exactly 2015 positive factors. What is the sum of the (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 claim 66 this is the smallest such integer. Note that N < 2 . 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. It 66 is easy to see that this means e + f > 66 so n > 2 > N . 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.