اگر $d=(a-۵\,,\,{{a}^{۲}}-۶a+۳)$ و $a\in \mathbb{Z}$ باشد، آنگاه $d$ کدام است؟ ($d$ بزرگترین مقسومعلیه است.)
$\left\{ \begin{matrix}d|a-5\Rightarrow d|(a-5)(a-1)={{a}^{2}}-6a+5\,\,\,\,\,\,\,\,(1) \\d|{{a}^{2}}-6a+3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\, \\\end{matrix} \right.\,\,\,\,$ $(1),(2)\Rightarrow d|({{a}^{2}}-6a+5)-({{a}^{2}}-6a+3)$ $\Rightarrow d|2\Rightarrow d=1\,$ یا $2$