PUMaC 2015 · 数论（B 组） · 第 3 题
PUMaC 2015 — Number Theory (Division B) — Problem 3
题目详情
- [ 4 ] What is the sum of all positive integers n such that lcm(2 n, n ) = 14 n − 24?
解析
- [ 4 ] What is the sum of all positive integers n such that lcm(2 n, n ) = 14 n − 24? Solution: 2 2 2 2 lcm(2 n, n ) is n when n is even and 2 n when n is odd. Solving the equation n = 14 n − 24, we 2 get n = 2 and n = 12, both of which are even and thus yield solutions. Solving 2 n = 14 n − 24, we get n = 3 and n = 4, of which only n = 3 works, because it is odd. Thus, our answer is 2 + 3 + 12 = 17 . Author: Eric Neyman