第1題
2N除以21餘數為20，除以23餘數為22，除以25餘數為24
……

第2題
\begin{align}   & {{n}^{12}}-{{n}^{8}}-{{n}^{4}}+1 \\ & ={{\left( n-1 \right)}^{2}}{{\left( n+1 \right)}^{2}}{{\left( {{n}^{2}}+1 \right)}^{2}}\left( {{n}^{4}}+1 \right) \\ \end{align}
n-1,n+1,{{n}^{2}}+1,{{n}^{4}}+1均為偶數，且n-1\ or\ n+1為4的倍數
故所求為9

如此罷了
\begin{align}   & {{n}^{12}}-{{n}^{8}}-{{n}^{4}}+1 \\ & ={{n}^{8}}\left( {{n}^{4}}-1 \right)-\left( {{n}^{4}}-1 \right) \\ & =\left( {{n}^{4}}-1 \right)\left( {{n}^{8}}-1 \right) \\ & ={{\left( {{n}^{4}}-1 \right)}^{2}}\left( {{n}^{4}}+1 \right) \\ & ={{\left( {{n}^{2}}-1 \right)}^{2}}{{\left( {{n}^{2}}+1 \right)}^{2}}\left( {{n}^{4}}+1 \right) \\ & ={{\left( n-1 \right)}^{2}}{{\left( n+1 \right)}^{2}}{{\left( {{n}^{2}}+1 \right)}^{2}}\left( {{n}^{4}}+1 \right) \\ \end{align}