mod性质 学习笔记

mod性质小结

\(a\equiv b(\mod m)\) $ \rightarrow \( \)a-b=k*m,k\in Z$

\(a\equiv b且c\equiv d(\mod m)\) \(\rightarrow\) \(a\pm c\equiv b\pm d(\mod m)\)

\(a\equiv b且c\equiv d(\mod m)\) \(\rightarrow\) \(ac\equiv bd(\mod m)\)

\(a\equiv b(\mod m)\) \(\rightarrow\) \(a^n\equiv b^n(\mod m),n>=0\)

\(ad\equiv bd(\mod m)\) \(\rightarrow\) \(a\equiv b(\mod \frac m{gcd(d,m)}),d\neq 0\)

\(a\equiv b(\mod md)\) \(\rightarrow\) \(a\equiv b(\mod m)\)

\(a\equiv b(\mod m)\)且\(a\equiv b(\mod n)\) \(\rightarrow\) \(a\equiv b(\mod lcm(n,m))\)

\(a\equiv b(\mod nm)\) \(\rightarrow\) \(a\equiv b(\mod m)\)且\(a\equiv b(\mod n),n\perp m\)
继续拆下去可以变成
\(a\equiv b(\mod m)\) \(\rightarrow\) \(a\equiv b(\mod p^{m_p})\)
其中\(p\)为\(m\)分解出来的质因数，\(m_p\)为该质因数有多少个