HMMT 二月 2017 · ALGNT 赛 · 第 4 题

HMMT February 2017 — ALGNT Round — Problem 4

Find all pairs ( a, b ) of positive integers such that a + b is a multiple of ab.

解析

Find all pairs ( a, b ) of positive integers such that a + b is a multiple of ab. Proposed by: Yang Liu 2017 Answer: (1 , 1) and (2 , 2 ) . 2017 We want ab | a + b. This gives that a | b . Therefore, we can set b = b a. Substituting this gives 2017 2 2017 2016 b a | a + b a , so b a | a + b . Once again, we get a | b , so we can set b = b a. 2017 2017 2017 2017 2017 2017 2016 i Continuing this way, if we have b a | a + b , then a | b , so we can set b = b a and derive i +1 i +1 i +1 i +1 i i − 1 2017 b a | a + b . Continuing down to i = 1, we would have b = b a so ab | 1 + b . If a ≥ 3, then i i 1 1 1 ab > 1 + b for all b ≥ 1, so we need either a = 1 or a = 2 . If a = 1, then b | b + 1, so b = 1 . This 1 1 1 2017 2017 k gives the pair (1 , 1) . If a = 2, we need 2 b | b + 2 . Therefore, we get b | 2 , so we can write b = 2 k +1 k 2017 2017 − k for 0 ≤ k ≤ 2017 . Then we need 2 | 2 + 2 . As k ≤ 2017, we need 2 | 1 + 2 . This can only 2017 happen is k = 2017 . This gives the pair (2 , 2 ).