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