Question

关于x的方程ax-x=2+a+2的解为非负整数求所有整数的和

Answer 1

将原方程化简得：ax - x - a = 2 + 2
移项得：(a - 1)x = a + 4
解得：x = (a + 4)/(a - 1)
因为x为非负整数，所以(a + 4)/(a - 1)为正整数，即(a + 4) mod (a - 1) = 0
(a + 4) mod (a - 1) = 0 等价于 (a + 4) - (a - 1) * [(a + 4) / (a - 1)] = 0
整理得：5 - (a - 1) * [(a + 4) / (a - 1)] = 0
化简得：(a + 9) / (a - 1) = 1 + 10 / (a - 1)
所以a - 1为10的因数，即a = 2, 3, 4, 5, 6, 7, 8, 11
将a分别代入原方程，解得x为6, 3, 2, 3/2, 4/3, 3/2, 6/5, 3，均为非负整数。
因此，所有整数的和为 6 + 3 + 2 + 3/2 + 4/3 + 3/2 + 6/5 + 3 = 29 5/15。