HMMT 十一月 2012 · 团队赛 · 第 1 题

英文解析

1. [ 3 ] Find the number of integers between 1 and 200 inclusive whose distinct prime divisors sum to 16.
   (For example, the sum of the distinct prime divisors of 12 is 2 + 3 = 5.)
   Answer: 6 The primes less than 16 are 2, 3, 5, 7, 11, and 13. We can write 16 as the sum of suchprimes in three different ways and find the integers less than 200 with those prime factors:
   • 13 + 3: 3 · 13 = 39 and 3 · 13 = 117.2
   • 11 + 5: 5 · 11 = 55
   2 2
   • 11 + 3 + 2: 2 · 3 · 11 = 66, 2 · 3 · 11 = 132, and 2 · 3 · 11 = 198.
   There are therefore 6 numbers less than 200 whose prime divisors sum to 16.