【数学A】例題4.2.3：方程式の整数解1（One More）★★

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% 例題A4.2.3：方程式の整数解1（One More）★★
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次の不定方程式の整数解を求めよ．

(1) $5x-4y=28$

(2) $45x+464y=14$

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(1) $5x-4y=28$より，$5x=4(y+7)\cdots(\mathrm{i})$

5と4は互いに素であるから，$x$は4の倍数となる．

したがって，$k$を整数として，$x=4k$とおける．

これを(i)に代入すると，$5\times 4k=4(y+7)$

$5k=y+7$より，$y=5k-7$

よって，求める整数解は，$x=4k,y=5k-7(k$は整数)

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$5x-4y=28$より，$y=\frac{5}{4}x-7$

$y$は整数より，$x$は4の倍数となる．

したがって，$x=4k(k$は整数)とおけ，$y=5k-7$

よって，求める整数解は，$x=4k,y=5k-7(k$は整数)

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(2) $464=45\times 10+14$より，

\[
45x+(45\times 10+14)y=14
\]

したがって，$45(x+10y)=14(1-y)\cdots(\mathrm{i})$

45と14は互いに素であるから，$x+10y$は14の倍数となる．

したがって，$k$を整数として，$x+10y=14k$，すなわち，$x=14k-10y\cdots(\mathrm{ii})$とおける．

(ii)を(i)に代入すると，$45\times 14k=14(1-y)$

$45k=1-y$より，$y=-45k+1$

これを(ii)に代入すると，$x=464k-10$

よって，求める整数解は，$x=464k-10,y=-45k+1(k$は整数)

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% 問題A4.2.3：方程式の整数解1（One More）★★
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次の不定方程式の整数解を求めよ．

(1) $7x-3y=18$

(2) $39x+56y=17$

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(1) $7x-3y=18$より，$7x=3(y+6)\cdots(\mathrm{i})$

7と3は互いに素であるから，$x$は3の倍数となる．

したがって，$k$を整数として，$x=3k$とおける．

これを(i)に代入すると，$7\times 3k=3(y+6)$

$7k=y+6$より，$y=7k-6$

よって，求める整数解は，$x=3k,y=7k-6(k$は整数)

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$7x-3y=18$より，$y=\frac{7}{3}x-6$

$y$は整数より，$x$は3の倍数となる．

したがって，$x=3k(k$は整数)とおけ，$y=7k-6$

よって，求める整数解は，$x=3k,y=7k-6(k$は整数)

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(2) $56=39\times 1+17$より，

\[
39x+(39\times 1+17)y=17
\]

したがって，$39(x+y)=17(1-y)\cdots(\mathrm{i})$

39と17は互いに素であるから，$x+y$は17の倍数となる．

したがって，$k$を整数として，$x+y=17k$，すなわち，$x=17k-y\cdots(\mathrm{ii})$とおける．

(ii)を(i)に代入すると，$39\times 17k=17(1-y)$

$39k=1-y$より，$y=-39k+1$

これを(ii)に代入すると，$x=56k-1$

よって，求める整数解は，$x=56k-1,y=-39k+1(k$は整数)

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% 側注1
4と28は1より大きい公約数4をもつ．

% 側注2
$x$が4の倍数ではないとき，$y$は整数ではない．

% 側注3
$x$と$y$の係数のうち，大きい数の464を小さい数の45で割ることを考える．

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% 側注1
3と18は1より大きい公約数3をもつ．

% 側注2
$x$が3の倍数ではないとき，$y$は整数ではない．

% 側注3
$x$と$y$の係数のうち，大きい数の56を小さい数の39で割ることを考える．

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(1) $5x-4y=28$を$5x=4(y+7)$と式変形してまとめ，5と4は互いに素であることを利用する．

(2) $x$と$y$の係数に注目すると，$464=45\times 10+14$という関係がある．式変形してまとめ，45と14は互いに素であることを利用する．

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不定方程式$ax+by=c$（$a$と$b$は互いに素）において，$a$または$b$が$c$と1より大きい公約数をもつときは，式変形してまとめ，互いに素であることを利用する．