問題の解答
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% 例題A4.1.10：整数の除法と余り（One More）★
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$a,b$を整数とする．$a$を5で割ると2余り，$b$を5で割ると3余る．このとき，次の数を5で割った余りを求めよ．
(1) $a-3b$
(2) $ab$
(3) $a^4$
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% 例題A4.1.10の解答（One More）★
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$a=5k+2,b=5l+3(k,l$は整数)と表される．
(1)
\begin{align*}
a-3b&=5k+2-3(5l+3)\\
&=5k+2-15l-9\\
&=5(k-3l-2)+3
\end{align*}
よって，求める余りは，3
(2)
\begin{align*}
ab&=(5k+2)(5l+3)\\
&=25kl+15k+10l+6\\
&=5(5kl+3k+2l+1)+1
\end{align*}
よって，求める余りは，1
(3)
\[
a^2=(5k+2)^2=25k^2+20k+4=5(5k^2+4k)+4
\]
したがって，$a^2=5m+4(m\text{は整数})$と表されるから，
\[
a^4=(a^2)^2=(5m+4)^2=25m^2+40m+16=5(5m^2+8m+3)+1
\]
よって，求める余りは，1
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% 問題A4.1.10：整数の除法と余り（One More）★
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$a,b$を整数とする．$a$を7で割ると4余り，$b$を7で割ると5余る．このとき，次の数を7で割った余りを求めよ．
(1) $a+2b$
(2) $ab$
(3) $a^4$
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% 問題A4.1.10の解答（One More）★
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$a=7k+4,b=7l+5(k,l$は整数)と表される．
(1)
\begin{align*}
a+2b&=7k+4+2(7l+5)\\
&=7k+4+14l+10\\
&=7(k+2l+2)
\end{align*}
よって，求める余りは，0
(2)
\begin{align*}
ab&=(7k+4)(7l+5)\\
&=49kl+35k+28l+20\\
&=7(7kl+5k+4l+2)+6
\end{align*}
よって，求める余りは，6
(3)
\[
a^2=(7k+4)^2=49k^2+56k+16=7(7k^2+8k+2)+2
\]
したがって，$a^2=7m+2(m\text{は整数})$と表されるから，
\[
a^4=(a^2)^2=(7m+2)^2=49m^2+28m+4=7(7m^2+4m)+4
\]
よって，求める余りは，4
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