こんにちは。相城です。最大公約数の問題です。(3)は互除法を使ってみよう。
次の数の最大公約数を求めなさい。
(1) 60, 90
(2) 18, 48, 60
(3) 1705, 8463
![](https://www.mathtext.info/blog/wordpress/wp-content/uploads/2020/02/1yohaku.png)
答え
(1)
![Rendered by QuickLaTeX.com 60=2^2\times3\times5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0e6d25ff9667efc89f0569ba484a2e3c_l3.png)
となり
共通する数字は2が1個, 3も1個, 5も1個だけ共通するので,
最大公約数は![Rendered by QuickLaTeX.com 2\times3\times5=30](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3225ed77b115996f26af2fe79b3c961a_l3.png)
(2)
![Rendered by QuickLaTeX.com 18=2\times3^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a5a36949b0e5ce8493056b55d89b8d08_l3.png)
![Rendered by QuickLaTeX.com 48=2^4\times3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-22e3c98e6cf141f6a8b4178388145a23_l3.png)
となり
共通する数字は2が1個と3も1個なので,
最大公約数は![Rendered by QuickLaTeX.com 2\times3=6](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-34d0ae9a4c925fec0a2719e5518e9407_l3.png)
(3)
![Rendered by QuickLaTeX.com 8463=1705\cdot4+1643](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c963adea5b4c82f160a0dce8c1401930_l3.png)
![Rendered by QuickLaTeX.com 1705=1643\cdot1+62](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-90a6c8959ffe253955257fd867e42bbf_l3.png)
![Rendered by QuickLaTeX.com 1643=62\cdot26+31](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-57dfc195d113c5156d51cbcae84901e2_l3.png)
![Rendered by QuickLaTeX.com 62=31\cdot2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9152653792a528f6837ca2a75fe16763_l3.png)
よって31
![Rendered by QuickLaTeX.com 60=2^2\times3\times5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0e6d25ff9667efc89f0569ba484a2e3c_l3.png)
![Rendered by QuickLaTeX.com 90=2\times3^2\times5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-47264cab84ad179a6006967243fa9537_l3.png)
共通する数字は2が1個, 3も1個, 5も1個だけ共通するので,
最大公約数は
![Rendered by QuickLaTeX.com 2\times3\times5=30](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3225ed77b115996f26af2fe79b3c961a_l3.png)
(2)
![Rendered by QuickLaTeX.com 18=2\times3^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a5a36949b0e5ce8493056b55d89b8d08_l3.png)
![Rendered by QuickLaTeX.com 48=2^4\times3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-22e3c98e6cf141f6a8b4178388145a23_l3.png)
![Rendered by QuickLaTeX.com 60=2^2\times\3\times5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-44b1a01300c882cb464de9c2a75c9b6c_l3.png)
共通する数字は2が1個と3も1個なので,
最大公約数は
![Rendered by QuickLaTeX.com 2\times3=6](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-34d0ae9a4c925fec0a2719e5518e9407_l3.png)
(3)
![Rendered by QuickLaTeX.com 8463=1705\cdot4+1643](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c963adea5b4c82f160a0dce8c1401930_l3.png)
![Rendered by QuickLaTeX.com 1705=1643\cdot1+62](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-90a6c8959ffe253955257fd867e42bbf_l3.png)
![Rendered by QuickLaTeX.com 1643=62\cdot26+31](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-57dfc195d113c5156d51cbcae84901e2_l3.png)
![Rendered by QuickLaTeX.com 62=31\cdot2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9152653792a528f6837ca2a75fe16763_l3.png)
よって31