4x-2-mod-9-7x-2-mod-13-

Question Number 175320 by cortano1 last updated on 27/Aug/22

{\begin{cases} 4 x = 2 (m o d 9) \\ 7 x = 2 (m o d 13) \end{cases}

Commented by cortano1 last updated on 27/Aug/22

i g o t x = 23 + 117 k s i r

Commented by Rasheed.Sindhi last updated on 27/Aug/22

s i r x = 23 + 117 k {d o e s n}^{'} t s a t i s f y

7 x \equiv 2 (m o d 13)

Commented by cortano1 last updated on 27/Aug/22

o y e s . i a m t y p o 6 \times 13 = 68

Answered by Rasheed.Sindhi last updated on 27/Aug/22

4 x \equiv 2 + 2 \times 9 (m o d 9)

x \equiv 5 (m o d 9)

x = 5 + 9 m

7 x \equiv 2 (m o d 13) \Rightarrow 7 (5 + 9 m) \equiv 2 (m o d 13

35 + 63 m \equiv 2 (m o d 13)

63 m \equiv - 33 (m o d 13)

m = 10

x = 5 + 9 m = 5 + 9 (10) = 95

x = 95 + k \times LCM (9, 13); k \in Z

x = 95 + 117 k; k \in Z

Commented by Tawa11 last updated on 27/Aug/22

Great sir

Answered by mr W last updated on 27/Aug/22

4 x = 9 k + 2

\Rightarrow x = 9 m + 5

7 (9 m + 5) = 13 h + 2

13 h - 63 m = 33

\Rightarrow m = 13 n + 10

\Rightarrow x = 9 (13 n + 10) + 5 = 117 n + 95

Answered by CElcedricjunior last updated on 28/Aug/22

{\begin{cases} 4 \boldsymbol x \equiv 2 [9] \\ 7 \boldsymbol x \equiv 2 [13] \end{cases} => {\begin{cases} 28 \boldsymbol x \equiv 14 [9] \\ 14 \boldsymbol x \equiv 4 [13] \end{cases} => {\begin{cases} \boldsymbol x \equiv 5 [9] \\ \boldsymbol x \equiv 4 [13] \end{cases}

=> \exists \boldsymbol p; \boldsymbol q \in Z / {\begin{cases} \boldsymbol x = 9 \boldsymbol p + 5 \\ \boldsymbol x = 13 \boldsymbol q + 4 \end{cases}

\boldsymbol o r \boldsymbol x = \boldsymbol x \Leftrightarrow 9 \boldsymbol p - 13 \boldsymbol q = - 1

\boldsymbol p g c d (9; 13) = 1 e t 1 / 1

\boldsymbol a l o r s \boldsymbol c e t t e \boldsymbol e q u a t i o n \boldsymbol a d m e t \boldsymbol d e \boldsymbol s o l u t i o n

\boldsymbol s o i t (- 3; - 2) \boldsymbol u n e \boldsymbol s o l u t i o n \boldsymbol p a r t i c u l i e e

9 \boldsymbol p - 13 \boldsymbol q = 9 (- 3) - 13 (- 2)

=> 9 (\boldsymbol p + 3) = 13 (\boldsymbol q + 2) (1)

=> 9 / 13 (\boldsymbol q + 2) \boldsymbol o r \boldsymbol p g c d (9; 13) = 1

=> 9 / \boldsymbol q + 2 \Leftrightarrow \exists \boldsymbol k \in Z / \boldsymbol q + 2 = 9 \boldsymbol k (2)

=> \boldsymbol q = 9 \boldsymbol k - 2

(2) \boldsymbol d a n s (1) => 9 (\boldsymbol p + 3) = 13 (9 \boldsymbol k)

=> \boldsymbol p + 3 = 13 \boldsymbol k => \boldsymbol p = 13 \boldsymbol k - 3

\boldsymbol d e \boldsymbol c e \boldsymbol q u i \boldsymbol p r e c \overset{`}{\boldsymbol e d e {\begin{cases} \boldsymbol x = 9 \boldsymbol p + 5 \\ \boldsymbol x = 13 \boldsymbol q + 4 \end{cases}}

\boldsymbol d o n c

\boldsymbol x = 9 (13 \boldsymbol k - 3) + 5 = 117 \boldsymbol k - 22

\boldsymbol o u

\boldsymbol x = 13 (9 \boldsymbol k - 2) + 4 = 117 \boldsymbol k - 22

\boldsymbol S_{Z} = {117 \boldsymbol k - 22 / \boldsymbol k \in Z}

S_{\boldsymbol I N} = {117 \boldsymbol k - 22 / \boldsymbol k \in] \frac{22}{117}; \to [}

\dots \dots \dots l e c \overset{´}{e l} \overset{`}{e b r e} c e d r i c j u n i o r \dots \dots \dots \dots