شاه کلید نکات
شاه کلید سوال
1 اگر a≠0 عددی صحیح و دو عدد (7m+6) و (6m+5) بر a بخش پذیر باشند ثابت کنید \(a = \pm 1\).
\(\begin{array}{l}\left. \begin{array}{l}a|7m + 6\; \Rightarrow \;a|42m + ...\\a|6m + 5\; \Rightarrow \;...|42m + ...\end{array} \right\}\;\\\\ \Rightarrow \;a|\left( {42m + 36} \right) - \left( {42m + 35} \right)\end{array}\)
(چرا؟) \( \Rightarrow \;\;a|1\;\; \Rightarrow \;\;a = \pm 1\)
\(\begin{array}{l}\left. \begin{array}{l}a|7m + 6\; \Rightarrow \;a|42m + 36\\a|6m + 5\; \Rightarrow \;a|42m + 35\end{array} \right\}\;\\\\ \Rightarrow \;a|\left( {42m + 36} \right) - \left( {42m + 35} \right)\\\\ \Rightarrow \;\;a|1\;\; \Rightarrow \;\;a = \pm 1\\\\\left| a \right| \le 1\;\;\;\mathop \Rightarrow \limits^{\left| a \right| \in \mathbb{N}} \;\;\;\left| a \right| = 1 \Rightarrow a = \pm 1\end{array}\)
2 اگر a|b نشان دهید که \({a^n}|{b^n}\).
\(a|b\; \Rightarrow \;b = aq\; \Rightarrow \;{b^n} = ...\;\mathop \Rightarrow \limits^{{q^n} = q'} \;{b^n} = ...\;q'\; \Rightarrow \;{a^n}|{b^n}\) :اثبات
\(a|b\; \Rightarrow \;b = aq\; \Rightarrow \;{b^n} = {a^n}\,{q^n}\;\mathop \Rightarrow \limits^{{q^n} = q'} \;{b^n} = {a^n}\,q'\; \Rightarrow \;{a^n}|{b^n}\)
3 اگر a|b و c|d نشان دهید که ac|bd.
\(\begin{array}{l}\left. \begin{array}{l}a|b\; \Rightarrow \;b = a{q_1}\\c|d\; \Rightarrow \;... = ...\end{array} \right\}\; \Rightarrow \;b \times d = \left( {a \times c} \right)\underbrace {\left( {{q_1}{q_2}} \right)}_q\\\\ \Rightarrow \;... = a \times c \times q\; \Rightarrow \;...|bd\end{array}\)
\(\begin{array}{l}\left. \begin{array}{l}a|b\; \Rightarrow \;b = a{q_1}\\c|d\; \Rightarrow \;d = c{q_2}\end{array} \right\}\; \Rightarrow \;b \times d = \left( {a \times c} \right)\underbrace {\left( {{q_1}{q_2}} \right)}_q\\\\ \Rightarrow \;bd = a \times c \times q\; \Rightarrow \;ac|bd\end{array}\)
4 اگر a|b و c|d نشان دهید که \(a|mb \pm nc\).
(از ویژگی 1 و ویژگی 3 استفاده کنید).
\(\left. \begin{array}{l}a|b \Rightarrow a|mb\\a|c \Rightarrow a|nc\end{array} \right\}\;\mathop \Rightarrow \limits^ \pm \;a|mb \pm nc\)
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