Solve-for-x-x-1-mod-3-x-4-mod-5-x-6-mod-7-

Question Number 115674 by naka3546 last updated on 27/Sep/20

S o l v e f o r x

x \equiv 1 m o d 3

x \equiv 4 m o d 5

x \equiv 6 m o d 7

Answered by floor(10²Eta[1]) last updated on 27/Sep/20

x \equiv 1 (\mod 3) \Rightarrow x = 3 a + 1

3 a + 1 \equiv 4 (\mod 5) \Rightarrow 3 a \equiv 3 (\mod 5) \Rightarrow a \equiv 1 (\mod 5)

a = 5 b + 1 \Rightarrow x = 3 (5 b + 1) + 1 = 15 b + 4

15 b + 4 \equiv 6 (\mod 7) \Rightarrow 15 b \equiv b \equiv 2 (\mod 7)

b = 7 c + 2 \Rightarrow x = 15 (7 c + 2) + 4

\Rightarrow x = 105 c + 34 \forall c \in Z