find-3x-4-mod-5-

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 {\dots, \frac{4}{3}, \frac{9}{3}, \frac{14}{3}, \dots}

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 {\dots . ., \frac{4}{3}, \frac{9}{3}, \frac{14}{3}, \dots .}

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)