3x-1-mod-4-4x-3-mod-5-5x-7-mod-11-

Question Number 127940 by bramlexs22 last updated on 03/Jan/21

{\begin{cases} 3 x = 1 (\mod 4) \\ 4 x = 3 (\mod 5) \\ 5 x = 7 (\mod 11) \end{cases}

Answered by floor(10²Eta[1]) last updated on 03/Jan/21

3 x \equiv 1 (\mod 4) \Rightarrow x \equiv 3 (\mod 4)

x = 4 a + 3

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

\Rightarrow a \equiv 1 (\mod 5) \Rightarrow a = 5 b + 1

\Rightarrow x = 4 (5 b + 1) + 3 = 20 b + 7

5 (20 b + 7) \equiv 7 (\mod 11)

\Rightarrow b \equiv 5 (\mod 11)

b = 11 c + 5 \Rightarrow x = 20 (11 c + 5) + 7

x = 220 c + 107, c \in Z

Answered by liberty last updated on 04/Jan/21

{\begin{cases} x = 3 (\mod 4) \dots (i) \\ x = 2 (\mod 5) \dots (ii) \\ x = 8 (\mod 11) \dots (iii) \end{cases}

for (i) \Rightarrow 55 a = 3 (\mod 4)

- a = 3 (\mod 4); a = - 3 (\mod 4)

for (ii) \Rightarrow 44 b = 2 (\mod 5)

4 b = 2 (\mod 5); b = 3 (\mod 5)

for (iii) \Rightarrow 20 c = 8 (\mod 11)

- 2 c = 8 (\mod 11); c = - 4 (\mod 11)

general solution

∴ 55 (- 3) + 44 (3) + 20 (- 4) + 220 k; k \in Z

i . e 220 k - 113 or 220 k + 107; k \in Z

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