HMMT 十一月 2012 · 团队赛 · 第 1 题

HMMT November 2012 — Team Round — Problem 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.)

解析

[ 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 such primes in three different ways and find the integers less than 200 with those prime factors: 2 • 13 + 3: 3 · 13 = 39 and 3 · 13 = 117. • 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.