${{x}^{2}}-8x+15\overset{4}{\mathop{\equiv }}\,0\Rightarrow (x-5)(x-3)\overset{4}{\mathop{\equiv }}\,0$ $1)\,x-5\overset{4}{\mathop{\equiv }}\,0\Rightarrow x\overset{4}{\mathop{\equiv }}\,5\overset{4}{\mathop{\equiv }}\,1\Rightarrow x=4k+1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(k\in \mathbb{Z})$ $2)\,x-3\overset{4}{\mathop{\equiv }}\,0\Rightarrow x\overset{4}{\mathop{\equiv }}\,3\Rightarrow x=4k+3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(k\in \mathbb{Z})$ $3)\,\left\{ \begin{align}  & x-3\overset{2}{\mathop{\equiv }}\,0\Rightarrow x\overset{2}{\mathop{\equiv }}\,3\overset{2}{\mathop{\equiv }}\,1 \\  & x-5\overset{2}{\mathop{\equiv }}\,0\Rightarrow x\overset{2}{\mathop{\equiv }}\,5\overset{2}{\mathop{\equiv }}\,1 \\ \end{align} \right.\Rightarrow x=2k+1\,\,\,\,\,\,(k\in \mathbb{Z})$