求方程123x+57y=531的整數解

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 between 0 and 9. Let y = v + 4.................(1)
When y = 0, x = 531/123 = 4.3, So x is between 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

因為x,y是整數

when x=1
177- 41 x1 = 136 不整除於19

when x=2
177-82 = 95 整除於19
所以x=2正確,將x=2代入 41x+19y=177
y=5

when x=3
177-3x41=177-123=54不整除於19

when x=4
177-4x41=13 不整除於１９



答案y=5 x=2

123(2) + 57(5)=531
246+285=531

正解！