some one write the statement a ≡−a(mod m) show that this statement is not generally true.! giving a counter example


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Question Number 64354 by Rio Michael last updated on 17/Jul/19

$s o m e o n e w r i t e t h e s t a t e m e n t$ $a \equiv - a (m o d m)$ $s h o w t h a t t h i s s t a t e m e n t i s n o t g e n e r a l l y t r u e .! g i v i n g a c o u n t e r$ $e x a m p l e$
	Answered by MJS last updated on 17/Jul/19

	$49 \div 11 =$ ${we}^{'} re looking for the greatest n with 11 n ⩽ 49 \Rightarrow$ $\Rightarrow n = 4$ $49 \div 11 = 4$ $multiplicating ‘ ‘ {backwards}^{″}$ $4 \times 11 = 44$ $subtracting$ $49 \div 11 = 4$ $- 44$ $===$ $5 remains$ $now do the same here :$ $(- 49) \div 11 =$ ${we}^{'} re looking for the greatest n with 11 n ⩽ (- 49) \Rightarrow$ $\Rightarrow n = (- 5)$ $(- 49) \div 11 = (- 5)$ $multiplicating ‘ ‘ {backwards}^{″}$ $(- 5) \times 11 = (- 55)$ $subtracting$ $(- 49) \div 11 = (- 5)$ $- (- 55)$ $=====$ $6 remains$ $but$ $49 \div (- 11) = (- 4)$ $- 44$ $===$ $5 remains$ $\Rightarrow for division with remainder :$ $(- a) : b \neq a : (- b)$ $remainders are always ⩾ 0 here$ $but some people use different logic which$ $is also ok . you have to follow the same$ $logic throughout your work$
	Commented by Rio Michael last updated on 17/Jul/19

	$t h a n k y o u s i r$
	Commented by Rio Michael last updated on 17/Jul/19

	$i f i d e c i d e t o t a k e t h e c o u n t e r e x a m p l e$ $7 \equiv - 7 (m o d 6)$ $a n d 6 ∤ 14 s o i t s a w r o n g s t a t e m e n t ?$
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