## 2016年10月23日日曜日

### 数学 – 代数学 - 整数 - 同値関係、合同式(剰余(mod))

$\begin{array}{l}a-b=ml\\ a-\left(b+mk\right)\\ =\left(a-b\right)-mk\\ =ml-mk\\ =m\left(l-k\right)\\ a\equiv b+mk\left(\mathrm{mod}5\right)\end{array}$

$\begin{array}{l}ac-bc=ml\\ \left(a-b\right)c=ml\\ \left(c,m\right)=1\\ a-b=mk\\ a\equiv b\left(\mathrm{mod}\text{\hspace{0.17em}}m\right)\end{array}$

$\begin{array}{l}ak-bk\\ =\left(a-b\right)k\\ =mlk\\ =lmk\\ ak\equiv bk\left(\mathrm{mod}\text{\hspace{0.17em}}mk\right)\end{array}$

$\begin{array}{l}a-b=mk\\ {a}_{1}d-{b}_{1}d={m}_{1}dk\\ {a}_{1}-{b}_{1}={m}_{1}k\\ {a}_{1}\equiv {b}_{1}\left(\mathrm{mod}\text{\hspace{0.17em}}{m}_{1}\right)\end{array}$

$\begin{array}{l}a-b=mk=m\text{'}dk=m\text{'}kd\\ a\equiv b\left(\mathrm{mod}\text{\hspace{0.17em}}d\right)\end{array}$

$\begin{array}{l}a\equiv b\left(\mathrm{mod}\text{\hspace{0.17em}}mm{\text{'}}_{1}\right),a\equiv b\left(\mathrm{mod}\text{\hspace{0.17em}}mm{\text{'}}_{k}\right)\\ a\equiv b\left(\mathrm{mod}\text{\hspace{0.17em}}m\right)\end{array}$

$\begin{array}{l}{\left(x+y\right)}^{p}-\left({x}^{p}+{y}^{p}\right)\\ =\sum _{r=0}^{p}\left(\begin{array}{c}p\\ r\end{array}\right){x}^{n-r}{y}^{r}-\left({x}^{p}+{y}^{p}\right)\\ ={x}^{p}+{y}^{p}+\sum _{r=1}^{p-1}\left(\begin{array}{c}p\\ r\end{array}\right){x}^{n-r}{y}^{r}-\left({x}^{p}+{y}^{p}\right)\\ =\sum _{r=1}^{p-1}\left(\begin{array}{c}p\\ r\end{array}\right){x}^{n-r}{y}^{r}\\ =pk\\ {\left(x+y\right)}^{p}\equiv {x}^{p}+{y}^{p}\left(\mathrm{mod}\text{\hspace{0.17em}}p\right)\end{array}$

$\begin{array}{l}{\left({x}_{1}+···+{x}_{n}+{x}_{n+1}\right)}^{p}\\ ={\left(\left({x}_{1}+···+{x}_{n}\right)+{x}_{n+1}\right)}^{p}\\ \equiv {\left({x}_{1}+···+{x}_{n}\right)}^{p}+{x}_{n+1}{}^{p}\\ \equiv {x}_{1}{}^{p}+···+{x}_{n}{}^{p}+{x}_{n+1}{}^{p}\left(\mathrm{mod}\text{\hspace{0.17em}}p\right)\end{array}$

(帰納法)