PUMaC 2017 · 团队赛 · 第 10 题
PUMaC 2017 — Team Round — Problem 10
题目详情
- (8) Given a positive integer x ≤ 233, let a be the remainder when x is divided by 233. Find the sum of all possible values of a .
解析
- Observe that 233 is prime, so a = 1 (mod 233) for any a . So x = ( x ) · x = 87 3 29 87 8 x = ( x ) . Furthermore, 29 · 8 = 232, ( x ) = 1 for all x , and the values of a correspond 8 to solutions to a = 1 (mod 233). For each solution, the negative of that solution is also a solution, so these 8 solutions come in four pairs, each of which sums to 233. Hence, the sum of all possible values of a is 233 · 4 = 932 . Problem written by Nathan Bergman