3x-2-mod-5-3x-4-mod-7-3x-6-mod-11-x-

Question Number 178747 by cortano1 last updated on 21/Oct/22

{\begin{cases} 3 x = 2 (\mod 5) \\ 3 x = 4 (\mod 7) \\ 3 x = 6 (\mod 11) \end{cases}

x = ?

Answered by Rasheed.Sindhi last updated on 21/Oct/22

{\begin{cases} 3 x = 2 (\mod 5) \dots . (i) \\ 3 x = 4 (\mod 7) \dots . (ii) \\ 3 x = 6 (\mod 11) \dots . (iii) \end{cases}; x = ?

(i) \Rightarrow 3 x \equiv 2 + 2 \times 5 (m o d 5)

\Rightarrow x \equiv 4 (m o d 5)

x = 5 k + 4

(i i) \Rightarrow 3 (5 k + 4) \equiv 4 (m o d 7)

15 k + 12 \equiv 4 (m o d 7)

15 k \equiv - 8 + 7 (14) (m o d 7)

k \equiv 6 (m o d 7)

k = 7 l + 6

x = 5 (7 l + 6) + 4 = 35 l + 34

(i i i) \Rightarrow 3 x \equiv 6 (m o d 11)

x \equiv 2 (m o d 11)

35 l + 34 \equiv 2 (m o d 11)

35 l \equiv - 32 (m o d 11)

35 l \equiv - 32 + 22 (11) (m o d 11)

35 l \equiv 210 (m o d 11)

l \equiv 6 (m o d 11)

l = 11 m + 6

x = 35 l + 34 = 35 (11 m + 6) + 34

= 385 m + 210 + 34 = 385 m + 244

x = 244, 385 m + 244

Commented by Tawa11 last updated on 21/Oct/22

Great sir

Commented by cortano1 last updated on 21/Oct/22

nice . thanks sir