こんにちは。相城です。
約数の基本問題を少し。定期テストに向けてご利用ください。
さて問題です。
(1) 20の正の約数をすべて求めよ。
(2) 36の約数をすべて求めよ。
(3) 72の正の約数の個数を求めよ。
(4) 400の正の約数の総和を求めよ。
(5) 12の倍数で正の約数の個数が10個である自然数を求めよ。
![](https://www.mathtext.info/blog/wordpress/wp-content/uploads/2020/02/1yohaku.png)
答え
(1) 1, 2, 4, 5, 10, 20
(2) ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36
(3)![Rendered by QuickLaTeX.com 72=2^3\times3^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-20cb7c2fb0ae495ad3431a335ae50c3a_l3.png)
(個)
(4)
より正の約数の総和は
![Rendered by QuickLaTeX.com (2^0+2^1+2^2+2^3+2^4)\times(5^0+5^1+5^2)=31\times31=961](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cbfcaa0346b4a9cf0cf72af2105ccb3e_l3.png)
961
(5)![Rendered by QuickLaTeX.com 12=2^2\times3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e93928d65d9805dac3f26e1a6e949652_l3.png)
正の約数の個数が10個ということは
とすると
なので
としたとき,
より
で
となること以外正の約数の個数が10個になることはないので, ![Rendered by QuickLaTeX.com a=4, b=1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9bfdd13a74408cc7d41d7718296bd647_l3.png)
したがって、![Rendered by QuickLaTeX.com 2^4\times3=48](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-15549ad800957aa2ba5a9b1afa645add_l3.png)
48
(2) ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36
(3)
![Rendered by QuickLaTeX.com 72=2^3\times3^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-20cb7c2fb0ae495ad3431a335ae50c3a_l3.png)
![Rendered by QuickLaTeX.com (3+1)\timse(2+1)=12](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-73f894f2de251ff6e39bce8f6a6b29f3_l3.png)
(4)
![Rendered by QuickLaTeX.com 400=2^4\times5^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ec554700c3ee07478b71101d14d94b7e_l3.png)
![Rendered by QuickLaTeX.com (2^0+2^1+2^2+2^3+2^4)\times(5^0+5^1+5^2)=31\times31=961](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cbfcaa0346b4a9cf0cf72af2105ccb3e_l3.png)
961
(5)
![Rendered by QuickLaTeX.com 12=2^2\times3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e93928d65d9805dac3f26e1a6e949652_l3.png)
正の約数の個数が10個ということは
![Rendered by QuickLaTeX.com 2^a\times3^b](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-20e9287fbd83daac847e3a37cc845dee_l3.png)
![Rendered by QuickLaTeX.com a\geqq2,\ b\geqq1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-1c5fe3fa58b4284395032b4aa07f0a6f_l3.png)
![Rendered by QuickLaTeX.com (a+1)\times (b+1)=10](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4477df89c08778cd012d802e2db9c80f_l3.png)
![Rendered by QuickLaTeX.com a\geqq2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-bf7cffdbcb27b3cc6d47355610a70f5a_l3.png)
![Rendered by QuickLaTeX.com a+1=5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-efbd1eada492c93ea1bffb5b99d933e7_l3.png)
![Rendered by QuickLaTeX.com b+1=2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-383da651e3bc31ade9f773ba308f1126_l3.png)
![Rendered by QuickLaTeX.com a=4, b=1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9bfdd13a74408cc7d41d7718296bd647_l3.png)
したがって、
![Rendered by QuickLaTeX.com 2^4\times3=48](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-15549ad800957aa2ba5a9b1afa645add_l3.png)
48