同余

概念

若两数除以\(c\)的余数相等，则称\(a,b\)膜\(c\)同余，记做\(a \equiv b (mod \ c)\)

同余方程的性质:

1.自反性:\(若a \equiv a (mod \ m)\)
2.对称性:\(若a \equiv b (mod \ m), 则b \equiv a (mod \ m)\)
3.传递性:\(若a \equiv b (mod \ m), b \equiv c (mod \ m), 则a \equiv c (mod \ m)\)
4.同加性:\(若a \equiv b (mod \ m),则a + c\equiv b+c (mod \ m)\)
5.同乘性:\(若a \equiv b (mod \ m),则若a \times c \equiv b \times c (mod \ m)\)
\(若a \equiv b (mod \ m), c \equiv d (mod \ m), 则a \times c\equiv b \times d (mod \ m)\)
6.同幂性:\(若a \equiv b (mod \ m), 则a^n \equiv b^n (mod \ m)\)
7.推论1：\(a \times b \ mod \ k = (a \ mod \ k) \times (b \ mod \ k) mod \ k\)
8.推论2：若$a \equiv b \ (mod \ c) \ d \mid c $ 则 \(a \equiv b \ (mod \ d)\)

例题（数学题）

试证明\(8888^{2222} + 7777^{3333} \equiv 0 \ (mod \ 37)\)

证明:
\(8888^{2222} + 7777^{3333}\)
\(\equiv 8^{2222} + 7^{3333} \ (mod \ 37)\)
\(\equiv 64^{1111} + 343^{1111} \ (mod \ 37)\)
\(\equiv (-10)^{1111} + 10^{1111} \ (mod \ 37)\)
\(\equiv 0 \ (mod \ 37)\)