find 3x≊4( mod 5)


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Question Number 146408 by tabata last updated on 13/Jul/21

$f i n d 3 x ≊ 4 (m o d 5)$
Commented by tabata last updated on 13/Jul/21

$h e l p m e s i r$
Commented by tabata last updated on 13/Jul/21

$? ? ? ? ? ? ? ? ?$
Commented by otchereabdullai@gmail.com last updated on 16/Jul/21

$nice question$
	Answered by gsk2684 last updated on 13/Jul/21

	$t o f i n d x s u c h t h a t 5 d i v i d e s 3 x - 4$ $3 x - 4 = 5 n$ $x = \frac{5 n + 4}{3}, n \in Z$ $x \in {. . ., \frac{4}{3}, \frac{9}{3}, \frac{14}{3}, . . .}$
	Commented by tabata last updated on 13/Jul/21

	$s i r c a n y o u g i v e m e s t e b b y s t e b p l e a s e$ $h o w x \in {. . . . ., \frac{4}{3}, \frac{9}{3}, \frac{14}{3}, . . . .}$
	Commented by gsk2684 last updated on 13/Jul/21

	$j u s t b y s u b s t i t i n g t h e v a l u e o f n$ $i f n = o t h e n x = \frac{5 (0) + 4}{3} = \frac{4}{3}$ $i f n = 1 t h e n x = \frac{5 (1) + 4}{3} = \frac{9}{3}$ $i f n = 2 t h e n x = \frac{5 (2) + 4}{3} = \frac{14}{3}$ $a n d s o o n .$
	Answered by Rasheed.Sindhi last updated on 13/Jul/21

	$3 x \equiv 4 (m o d 5)$ $3 x \equiv 4 + 5 (m o d 5)$ $3 x \equiv 9 (m o d 5)$ $x \equiv 3 (m o d 5)$ $x = 3 + 5 k; \forall k \in Z$
	Answered by physicstutes last updated on 13/Jul/21

	$3 x \equiv 4 (\mod 5) \Rightarrow 3 x = 5 y + 4 where y \in Z$ $so we solve the linear diophantine equation$ $3 x - 5 y = 4$ $5 = 1 (3) + 2$ $3 = 1 (2) + 1$ $2 = 2 (1) + 0$ $Reversing the steps in euclids algorithm just done above,$ $1 = 3 - 1 (2)$ $= 3 - 1 [5 - 1 (3)]$ $= 3 - 1 (5) + 1 (3)$ $= 2 (3) - 1 (5)$ $= 3 [5 - 1 (3)] - 1 (5)$ $= - 3 (3) - 2 (5)$ $∵ x = 3 is a good solution other solutions are such that$ $x \equiv 3 (\mod 5)$
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