こんにちは。相城です。ユーグリッドの互除法を使って求めるパターンを置いておきます。それではどうぞ。
(1) を満たす整数
の組を1つ見つけよ。
(2) を満たすすべての整数解を求めよ。
(3) を満たす整数
の組を1つ見つけよ。
(4) を満たす整数
の組を1つ見つけよ。
![](https://mathtext.info/blog/wp-content/uploads/2020/02/1yohaku.png)
答え
(1)
より
・・・①
より
・・・②
より
・・・③
②を③にあてはめて,
![Rendered by QuickLaTeX.com 1=7-3(9-7\cdot1)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-dbf830e326d8bda8d74e536129b93a17_l3.png)
![Rendered by QuickLaTeX.com 1=7\cdot4-9\cdot3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-3e793b42ed8cfa93b3765a746f79e4e3_l3.png)
これに①をあてはめて,
![Rendered by QuickLaTeX.com 1=4(34-9\cdot3)-9\cdot3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8d85faa5f10a25638e5d323ddd3c41a3_l3.png)
![Rendered by QuickLaTeX.com 1=34\cdot4-9\cdot15](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-ed80539c597e7c38becc04431856bf44_l3.png)
よって![Rendered by QuickLaTeX.com (x, y)=(4, 15)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c11f277289706c77ee9b30561a4dbfec_l3.png)
(2)
と
を筆算でひくと,
となり、34と9は互いに素だから,
とおける。
よって求める整数解は、
(
は整数)
となる。
(3)
(1)より、(1)の答えを5倍したものが答え。よって, この場合,
が答え。
(4)
まず
を満たす整数
の組を1組見つけて3倍することを考える。
より
・・・①
より
・・・②
より
・・・③
より
・・・④
④に③をあてはめて,
![Rendered by QuickLaTeX.com 1=3-(5-3\cdot1)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-65d7bf44bd4655f289a6e5275228b665_l3.png)
![Rendered by QuickLaTeX.com 1=-5+3\cdot2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-b4cd9fbce1bff2fce41454d6cc3bd57e_l3.png)
これに②をあてはめて,
![Rendered by QuickLaTeX.com 1=-5+2(13-5\cdot2)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-9b3160289855d539b9e6789906abb48f_l3.png)
![Rendered by QuickLaTeX.com 1=13\cdot2-5\cdot5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c8e1484cafd79cfa539d863e18169556_l3.png)
これに①をあてはめて,
![Rendered by QuickLaTeX.com 1=13\cdot2-5(31-13\cdot2)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-315dfc7b96419e28cd3db39d07258d55_l3.png)
![Rendered by QuickLaTeX.com 1=13\cdot12+31\cdot(-5)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f39d65f4ea60f7105b6afbfd025b7e7f_l3.png)
よって
であるから,
を満たす整数
の1組は![Rendered by QuickLaTeX.com (x, y)=(36, -15)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f3be14e512189ce423761f1e7289760a_l3.png)
![Rendered by QuickLaTeX.com 34=9\cdot3+7](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e81b5fced31d67258ec52ff0ff3eadde_l3.png)
![Rendered by QuickLaTeX.com 7=34-9\cdot3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-3434e8864275d6de93eeccf9ec2333bb_l3.png)
![Rendered by QuickLaTeX.com 9=7\cdot1+2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-ce8c5efa655de6a16cdc9dd9eec68102_l3.png)
![Rendered by QuickLaTeX.com 2=9-7\cdot1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2fd9689786b7a7f33ddc99e8e904cb0c_l3.png)
![Rendered by QuickLaTeX.com 7=2\cdot3+1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-09bb66eb4ac9a86d4dc9ad5e47bfd0e8_l3.png)
![Rendered by QuickLaTeX.com 1=7-2\cdot3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a39af3b39f2394232b4301e0faf10811_l3.png)
②を③にあてはめて,
![Rendered by QuickLaTeX.com 1=7-3(9-7\cdot1)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-dbf830e326d8bda8d74e536129b93a17_l3.png)
![Rendered by QuickLaTeX.com 1=7\cdot4-9\cdot3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-3e793b42ed8cfa93b3765a746f79e4e3_l3.png)
これに①をあてはめて,
![Rendered by QuickLaTeX.com 1=4(34-9\cdot3)-9\cdot3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8d85faa5f10a25638e5d323ddd3c41a3_l3.png)
![Rendered by QuickLaTeX.com 1=34\cdot4-9\cdot15](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-ed80539c597e7c38becc04431856bf44_l3.png)
よって
![Rendered by QuickLaTeX.com (x, y)=(4, 15)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c11f277289706c77ee9b30561a4dbfec_l3.png)
(2)
![Rendered by QuickLaTeX.com 34x-9y=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2fb774dab9d1a233962de5de4593e3d3_l3.png)
![Rendered by QuickLaTeX.com 34\cdot4-9\cdot15=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-aaf7604d0e15cc6dfb74d4d4804b7cf7_l3.png)
![Rendered by QuickLaTeX.com 34(x-4)-9(y-15)=0](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c466aa4cf9e1eaf2872f768eadc4eb7a_l3.png)
![Rendered by QuickLaTeX.com x-4=9k,\ y-15=34k](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c6647f8b631e4e96f651e826edd4e385_l3.png)
よって求める整数解は、
![Rendered by QuickLaTeX.com x=9k+4,\ y=34k+15](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a7364a0df81ea8673adbe79a4fd11419_l3.png)
![Rendered by QuickLaTeX.com k](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-3e55731b2ccdac9bcef26d20d0e6fd79_l3.png)
となる。
(3)
(1)より、(1)の答えを5倍したものが答え。よって, この場合,
![Rendered by QuickLaTeX.com (x, y)=(20, 75)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-18d0710e3be6be16b1cce2200ddeae40_l3.png)
(4)
まず
![Rendered by QuickLaTeX.com 13x+31y=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f7923f439f24d0738aeeb603b0865493_l3.png)
![Rendered by QuickLaTeX.com x, y](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-68ebd4e79d64956a644280148c2f9c3c_l3.png)
![Rendered by QuickLaTeX.com 31=13\cdot2+5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8c45b36eba5df1225ec93bf052c73f5c_l3.png)
![Rendered by QuickLaTeX.com 5=31-13\cdot2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-3a705d39d87ef242be1b40cedb50ff3c_l3.png)
![Rendered by QuickLaTeX.com 13=5\cdot2+3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-00ebc8776d3acc07d2944ebe6bd23f11_l3.png)
![Rendered by QuickLaTeX.com 3=13-5\cdot2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-df1dba7c4b9f51f3d988931fd6fa2514_l3.png)
![Rendered by QuickLaTeX.com 5=3\cdot1+2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-82df67658dfa1c29a9b8393e96b253a1_l3.png)
![Rendered by QuickLaTeX.com 2=5-3\cdot1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-390bc2e231ea056e9d46b8d4a2989983_l3.png)
![Rendered by QuickLaTeX.com 3=2\cdot1+1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2c714cb4e0433d03f042b52260f95c27_l3.png)
![Rendered by QuickLaTeX.com 1=3-2\cdot1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-74c8f8e8ab9bca6086bb3b5f76aa92d7_l3.png)
④に③をあてはめて,
![Rendered by QuickLaTeX.com 1=3-(5-3\cdot1)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-65d7bf44bd4655f289a6e5275228b665_l3.png)
![Rendered by QuickLaTeX.com 1=-5+3\cdot2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-b4cd9fbce1bff2fce41454d6cc3bd57e_l3.png)
これに②をあてはめて,
![Rendered by QuickLaTeX.com 1=-5+2(13-5\cdot2)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-9b3160289855d539b9e6789906abb48f_l3.png)
![Rendered by QuickLaTeX.com 1=13\cdot2-5\cdot5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c8e1484cafd79cfa539d863e18169556_l3.png)
これに①をあてはめて,
![Rendered by QuickLaTeX.com 1=13\cdot2-5(31-13\cdot2)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-315dfc7b96419e28cd3db39d07258d55_l3.png)
![Rendered by QuickLaTeX.com 1=13\cdot12+31\cdot(-5)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f39d65f4ea60f7105b6afbfd025b7e7f_l3.png)
よって
![Rendered by QuickLaTeX.com (x, y)=(12, -5)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f313fc479e4dcf369ab01db81509880a_l3.png)
![Rendered by QuickLaTeX.com 13x+31y=3](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-459c305ef9911d30e14fd3091d14d459_l3.png)
![Rendered by QuickLaTeX.com x, y](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-68ebd4e79d64956a644280148c2f9c3c_l3.png)
![Rendered by QuickLaTeX.com (x, y)=(36, -15)](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f3be14e512189ce423761f1e7289760a_l3.png)