整数約数と倍数最大公約数、最小公倍数、結合律

学習環境

親切な代数学演習新装2版―整数・群・環・体 (加藤明史(著)、現代数学社)の第Ⅰ部(整数)、第1章(約数と倍数)の問22の解答を求めてみる。

a、 b 、 c の最大公約数をgとする。

(a, b, c) = g

このとき、

\begin{array}{l} a = a_{0} g \\ b = b_{0} g \\ c = c_{0} g \\ (a_{0}, b_{0}, c_{0}) = 1 \end{array}

よって、

\begin{array}{l} ((a, b), c) \\ = ((a_{0} g, b_{0} g), c_{0} g) \\ = ((a_{0}, b_{0}) g, c_{0} g) \\ = ((a_{0}, b_{0}), c_{0}) g \\ = (a_{0}, b_{0}, c_{0}) g \\ = g \\ (a, (b, c)) \\ = (a_{0} g, (b_{0} g, c_{0} g)) \\ = (a_{0} g, (b_{0}, c_{0}) g) \\ = (a_{0}, (b_{0}, c_{0})) g \\ = (a_{0}, b_{0}, c_{0}) g \\ = 1 \end{array}

ゆえに、

((a, b), c) = (a, (b, c)) = (a, b, c)

（証明終）

\begin{array}{l} [[a, b], c] = [[a_{0} g, b_{0} g], c_{0} g] \\ = [[a_{0}, b_{0}] g, c_{0} g] \\ = [[a_{0}, b_{0}], c_{0}] g \\ = [a_{0}, b_{0}, c_{0}] g \\ = a_{0} b_{0} c_{0} g \\ = [a, b, c] \\ [a, [b, c]] \\ = [a_{0} g, [b_{0}, c_{0}] g] \\ = [a_{0}, [b_{0}, c_{0}]] g \\ = [a_{0}, b_{0}, c_{0}] g \\ = a_{0} b_{0} c_{0} g \\ = [a, b, c] \end{array}

よって、

\begin{array}{l} [[a, b], c] \\ = [a, [b, c]] \\ = [a, b, c] \end{array}

（証明終）