こんにちは。相城です。佐賀県の2019年度入試問題より、素数と方程式の問題をお届けします。それではどうぞ。
(ア)~(ウ)の問いに答えなさい。
(ア) 10以下の素数をすべて書きなさい。
(イ) 次の自然数の中から素数をすべて選び、書きなさい。
(ウ) ある素数
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
ただし。、
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![](https://www.mathtext.info/blog/wordpress/wp-content/uploads/2020/02/1yohaku.png)
答え
(ア)
![Rendered by QuickLaTeX.com 2,\ 3,\ 5,\ 7](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a84158f7e7d8334f3aafae5382e52806_l3.png)
(イ)
![Rendered by QuickLaTeX.com 23,\ 37,\ 71](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4aec34cf87216dc164290991f8d3a1c6_l3.png)
(ウ)
求める素数を
とすると、
![Rendered by QuickLaTeX.com x^2+52=17x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0c5d37353b938e58508c0c15d5df3b95_l3.png)
![Rendered by QuickLaTeX.com x^2-17x+52=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3e4c9d251422dfbdce7185dd7ba3d6c8_l3.png)
この二次方程式を解くと
![Rendered by QuickLaTeX.com (x-13)(x-4)=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8dcae0033189c4d9d13894316062a2b4_l3.png)
![Rendered by QuickLaTeX.com x=13,4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-97bfdd6c86b66c68ac9b8ff6486a99ca_l3.png)
は素数なので
は不適
よって![Rendered by QuickLaTeX.com x=13](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d3c8fe5fe542f8e7695c49e95ed28df3_l3.png)
求める素数は13
![Rendered by QuickLaTeX.com 2,\ 3,\ 5,\ 7](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a84158f7e7d8334f3aafae5382e52806_l3.png)
(イ)
![Rendered by QuickLaTeX.com 23,\ 37,\ 71](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4aec34cf87216dc164290991f8d3a1c6_l3.png)
(ウ)
求める素数を
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com x^2+52=17x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0c5d37353b938e58508c0c15d5df3b95_l3.png)
![Rendered by QuickLaTeX.com x^2-17x+52=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3e4c9d251422dfbdce7185dd7ba3d6c8_l3.png)
この二次方程式を解くと
![Rendered by QuickLaTeX.com (x-13)(x-4)=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8dcae0033189c4d9d13894316062a2b4_l3.png)
![Rendered by QuickLaTeX.com x=13,4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-97bfdd6c86b66c68ac9b8ff6486a99ca_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com x=4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-dd20e30eb7090a2f6b2b83ae682de63d_l3.png)
よって
![Rendered by QuickLaTeX.com x=13](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d3c8fe5fe542f8e7695c49e95ed28df3_l3.png)
求める素数は13