PUMaC 2010 · 数论（B 组） · 第 5 题

PUMaC 2010 — Number Theory (Division B) — Problem 5

Given that x , y , and z are positive integers such that + + = 2. Find the sum of all y z x possible x values.

解析

Given that x , y , and z are positive integers and all relatively prime such that + + = 2. y z x Find the number of all possible x values. y x z Solution: Write = p , = q and = r . Then we have p + q + r = 2 and pqr = 1. If one y z x of p , q , or r were 1, another would be at least one and the third would be positive, for a sum that is greater than 2. If two were greater than one, the sum would be greater than 2. Thus, 1 two are less than 1. If either of them were less than , then the third would be greater than 2 1