Question

求方程123x+57y=531的整数解

Answer 1

更新1:

Let y = v + 4.................(1) Let x = u + 2........(2) 呢到点黎

更新2:

Let y = v + 4.................(1) Let x = u + 2........(2) 点黎

To find the integral solution for 123x + 57y = 531
first we reduce the equation to: Before finding the general solution
we need to find one simple solution first. by guess
we have 41(-1) + 19(2) = 3 and hence
41(-59) + 19(118) = 177 so one such solution is (-59
118) 41x + 19y = 177 Since LCM (41
19) = 779
so 41(-59) + 779k - 779k + 19(118) = 177
where k is an integer 41(-59 + 19k) + 19(118 - 41k) = 177 So the general integral solution is x = -59 + 19k
y = 118 - 41k
where k is an integer For example
take k = 0
we get the solution we guessed before.
When x = 0
y = 531/57 = 9.3. So for both x and y to be positive integers
y must be beeen 0 and 9. Let y = v + 4.................(1) When y = 0
x = 531/123 = 4.3
So x is beeen 0 and 4. Let x = u + 2........(2) Sub. (1) and (2) into the equation
we get 123(u + 2) + 57(v + 4) = 531 123u + 246 + 57v + 228 = 531 123u + 57v = 57 so u = 0 and v = 1. Therefore
x = 0+ 2 = 2 and y = 1 +4 = 5.
123x+57y=531 一起除3 41x+19y=177 因为x
y是整数 when x=1 177- 41 x1 = 136 不整除于19 when x=2 177-82 = 95 整除于19 所以x=2正确
将x=2代入 41x+19y=177 y=5 when x=3 177-3x41=177-123=54不整除于19 when x=4 177-4x41=13 不整除于19 答案y=5 x=2 123(2) + 57(5)=531 246+285=531 正解！
参考: me