HMMT 十一月 2017 · GEN 赛 · 第 1 题
HMMT November 2017 — GEN Round — Problem 1
题目详情
- Find the sum of all positive integers whose largest proper divisor is 55. (A proper divisor of n is a divisor that is strictly less than n .)
解析
- Find the sum of all positive integers whose largest proper divisor is 55. (A proper divisor of n is a divisor that is strictly less than n .) Proposed by: Michael Tang Answer: 550 n The largest proper divisor of an integer n is , where p is the smallest prime divisor of n . So n = 55 p p for some prime p . Since 55 = 5 · 11, we must have p ≤ 5, so p = 2 , 3 , 5 gives all solutions. The sum of these solutions is 55(2 + 3 + 5) = 550.