求85x-74y=12所有的整數解，求滿足0 < x < 100的整數解。?

85x - 74y = 12
y = (85x - 12)/74 = x + (11x - 12)/74
(11x - 12)/74 = y - x = a
x = (74a + 12)/11
x = 6a + (8a + 12)/11
(8a + 12)/11 = x - 6a = b
a = (11b - 12)/8 = b + (3b - 12)/8
(3b - 12)/8 = a - b = c
b = (8c + 12)/3 = (8/3)c + 4
令 c = 3k , 則 b = 8k+4 , a = 11k+4 , x = 6(11k+4)+8k+4 = 74k + 28 , y = 74k+28 + 11k+4 = 85k + 32.

85x - 74y = 12 所有的整數解為 x = 74k + 28 , y = 85k + 32.

0 < 74k + 28 < 100
k = 0 , 滿足0 < x < 100的整數解為 x = 28.

別解:
85x ≡ 12 (mod74) ... ①
74x ≡ 0 (mod74) ... ②
① - ②:
11x ≡ 12 (mod74) ... ③
③×7 - ②:
3x ≡ 84 ≡ 10 (mod74) ... ④
④×4 - ③:
x ≡ 28 (mod74)
x = 28 + 74k
0 < 28 + 74k < 100
k = 0
x = 28