PUMaC 2012 · 数论（A 组） · 第 7 题

PUMaC 2012 — Number Theory (Division A) — Problem 7

[ 7 ] Let a , b , and c be positive integers satisfying 4 2 2 4 a + a b + b = 9633 2 2 2 2 5 2 a + a b + 2 b + c = 3605 . What is the sum of all distinct values of a + b + c ?

解析

[ 7 ] Let a , b , and c be positive integers satisfying 4 2 2 4 a + a b + b = 9633 2 2 2 2 5 2 a + a b + 2 b + c = 3605 . What is the sum of all distinct values of a + b + c ? Solution: We begin by summing the two systems and adding 1 to each side to obtain 4 2 2 4 2 2 5 a + 2 a b + b + 2 a + 2 b + 1 + c = 13239 , which we can rewrite as 3 2 2 2 5 ( a + b + 1) + c = 13239 . Now we consider this system modulo 11, because the least common multiple of 2 and 5 is 10, 10 and by Fermat’s Little Theorem, we have x ≡ 1(mod 11) whenever x is not a multiple of ( ) 2 5 5