PUMaC 2014 · 团队赛 · 第 6 题

PUMaC 2014 — Team Round — Problem 6

[ 6 ] Find the sum of positive integer solutions of x for = p , where p is 1716 x a prime. (If there are no solution, answer 0.)

解析

[ 6 ] Find the sum of positive integer solutions of x for = p , where p is a prime. (If 1716 − x there are no solution, answer 0.) Solution: 2 We get that x + xp − 1716 p = 0, so x x = − 1716 p and x + x = − p . We know that 1 2 1 2 2 1716 = 2 × 3 × 11 × 13, and also x , x are both divisible by p . Therefore, we try p = 2 , 3 , 11 , 13 1 2 and get two solutions for p : when p = 11, x = 132, x = − 143 and when p = 13, x = 143, 1 2 1 x = − 156. Thus, the sum of positive values of x are 132 + 143 = 275 . 2