## 2016年10月29日土曜日

### 数学 – 代数学 - 整数 - 1次の合同式(連立合同式の解法)

$\begin{array}{l}m=3·7·16=336\\ {M}_{1}=112,{M}_{2}=48,{M}_{3}=21\\ 112{t}_{1}\equiv 1\left(\mathrm{mod}\text{\hspace{0.17em}}3\right),48{t}_{2}\equiv 1\left(\mathrm{mod}\text{\hspace{0.17em}}7\right),21{t}_{3}\equiv 1\left(\mathrm{mod}\text{\hspace{0.17em}}16\right)\\ 112\equiv 1\left(\mathrm{mod}\text{\hspace{0.17em}}3\right)\\ {t}_{1}\equiv 1\left(\mathrm{mod}\text{\hspace{0.17em}}3\right),-{t}_{2}\equiv 1\left(\mathrm{mod}\text{\hspace{0.17em}}7\right),5{t}_{3}\equiv 1\left(\mathrm{mod}\text{\hspace{0.17em}}16\right)\\ 15{t}_{3}\equiv 3\left(\mathrm{mod}\text{\hspace{0.17em}}16\right)\\ 15\equiv -1\left(\mathrm{mod}\text{\hspace{0.17em}}16\right)\\ -{t}_{3}\equiv 3\left(\mathrm{mod}\text{\hspace{0.17em}}16\right)\\ {t}_{1}=1,{t}_{2}=-1,{t}_{3}=-3\\ x\equiv 112·{b}_{1}+48\left(-1\right){b}_{2}+21\left(-3\right){b}_{3}\\ \equiv 112{b}_{1}-48{b}_{2}-63{b}_{3}\left(\mathrm{mod}\text{\hspace{0.17em}}336\right)\\ {x}_{0}\equiv 112·1-48·4-63·11\\ \equiv 112-192-693\\ \equiv -773\\ \equiv -773+336·3\\ \equiv 235\left(\mathrm{mod}\text{\hspace{0.17em}}336\right)\end{array}$