Solve the system of congruences x ≡ 2 (mod 3) x ≡ 5( mod 7)


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Question Number 80504 by Rio Michael last updated on 03/Feb/20

$Solve the system of congruences$ $x \equiv 2 (\mod 3)$ $x \equiv 5 (\mod 7)$
Commented by mr W last updated on 03/Feb/20

$x = 21 n + 5$
Commented by Rio Michael last updated on 03/Feb/20

$p l e a s e c h e c k t h i s s i r :$ $x \equiv 2 (m o d 3) \Rightarrow x = 3 k + 2$ $3 k + 2 \equiv 5 (m o d 7) \Rightarrow 3 k \equiv 3 (m o d 7) k h a s n o$ $s o l u t i o n h e r e s i r!$ $f o r c h i n e s e r e m a i n d e r t h e o r e m$ $R_{1} = 7 a n d R_{2} = 3$ $7 X_{1} \equiv 1 (m o d 3) \Rightarrow X_{1} \equiv 1 (m o d 3)$ $3 X_{2} \equiv 1 (m o d 7) \Rightarrow X_{2} \equiv 5 (m o d 7)$ $x \equiv R_{1} a_{1} X_{1} + R_{2} a_{2} X_{2} (m o d n_{1} n_{2})$ $x \equiv (7) (2) (1) + (3) (5) (5) (m o d 21)$ $x \equiv 5 (m o d 21)$
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