同余式的基本性质

1.自反性：\(a\equiv a(\bmod m)\)

2.对称性：若 \(a\equiv b(\bmod m)\) ,则 \(b\equiv a(\bmod m)\)

3.传递性：若 \(a\equiv b(\bmod m)\) ,\(b\equiv c(\bmod m)\),则 \(a\equiv c(\bmod m)\)

4.消去性：$ac\equiv bc(\bmod p ) \to a \equiv b(\bmod \frac{p}{gcd(c,p)}) $

5.\(a\equiv b(\bmod cd \to a\equiv b(\bmod d)\)

6.\((a\equiv b(\bmod d),a\equiv b(\bmod c) \to a\equiv b(\bmod lcm(c,d))\)

7. 若 \(a\equiv b(\bmod p)\),则对任意 \(c\)有\((a+c)\equiv (b+c)(\bmod p)\)

8. 若 \(a\equiv b(\bmod p)\),则对任意 \(c\)有\((a\times c)\equiv (b\times c)(\bmod p)\)

9. 若 \(a\equiv b(\bmod p)\),则对任意 \(c\)有\((a^c)\equiv (b^c)(\bmod p)\)

10. 若 \(a\equiv b(\bmod p)\),则 \(c\)有\((a+c)\equiv (b+d)(\bmod p)\)、\((a-c)\equiv (b-d)(\bmod p)\) 和 \((a\times c)\equiv (b\times d)(\bmod p)\)