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Question Number 63499 by Rio Michael last updated on 04/Jul/19

given that   a∣b, show that −a∣b.

$${given}\:{that}\:\:\:{a}\mid{b},\:{show}\:{that}\:−{a}\mid{b}. \\ $$

Answered by MJS last updated on 04/Jul/19

a∣b ⇒ (b/a)=c with c∈Z  ⇒ −(b/a)=(−1)(b/a)=(−1)c=−c  c∈Z ⇔ −c∈Z

$${a}\mid{b}\:\Rightarrow\:\frac{{b}}{{a}}={c}\:\mathrm{with}\:{c}\in\mathbb{Z} \\ $$$$\Rightarrow\:−\frac{{b}}{{a}}=\left(−\mathrm{1}\right)\frac{{b}}{{a}}=\left(−\mathrm{1}\right){c}=−{c} \\ $$$${c}\in\mathbb{Z}\:\Leftrightarrow\:−{c}\in\mathbb{Z} \\ $$

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