PUMaC 2011 · 数论（A 组） · 第 4 题
PUMaC 2011 — Number Theory (Division A) — Problem 4
题目详情
- [ 4 ] For how many ordered triplets of three positive integers is it true that their product is four more than twice their sum?
解析
- Let the set be { a, b, c } , and without loss of generality, suppose that a ≤ b ≤ c . Then abc − 2 a − 2 b − 2 c = 4 . If c ≥ b ≥ a ≥ 4, then 4 = abc − 2 a − 2 b − 2 c ≥ 16 c − 2 c − 2 c − 2 c = 10 c ≥ 40 , which is a contradiction. Thus, a ∈ { 1 , 2 , 3 } . If a = 1, we get that bc − 2 b − 2 c = 6. Completing the rectangle, bc − 2 b − 2 c +4 = ( b − 2)( c − 2) =