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I've started to study number theory, I completely do not understand from my notes how to work this out. Could anyone show me with simple example how to solve this?

\begin{cases} 3x \equiv 4 \pmod{7}\\ 5x \equiv 9 \pmod{11} \end{cases}

Bill Dubuque
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etoRatio
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    Solve separately each congruence, then use the inverse isomorphism of the Chinese remainder theorem to determine the congruence class of $x\bmod 7\cdot 11$. – Bernard Jul 26 '21 at 21:01
  • Apply the theorem in the linked dupe to scale the congruences to an equivalent system where the coefficients on $x$ are $\equiv 1,,$ then apply CRT to solve that system. – Bill Dubuque Jul 26 '21 at 21:46
  • By here $,\bmod 7!:\ x\equiv 4/3\equiv -3/3\equiv -1,,$ and $\bmod 11!:\ x\equiv 9/5\equiv 20/5\equiv 4.,$ Next apply Easy CRT to solve $,x\equiv -1\pmod{7},\ x\equiv 4\pmod{11}\ \ $ – Bill Dubuque Jul 26 '21 at 21:52

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$$3x\equiv 4 \mod 7$$

$$5\times 3x\equiv 4 \times 5 \mod 7$$

$$x\equiv 6 \mod 7$$

$x=7t+6, t\in Z$

$$5x\equiv 9 \mod 11$$

$$35t+30\equiv 9 \mod 11$$

$$2t\equiv 1 \mod 11$$

$$t\equiv 6 \mod 11$$

$t=11k+6, k\in Z$

$$x=77k+48, k\in Z $$

Lion Heart
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