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I am unable to use the Chinese Remainder Theorem when the modulus is not coprime. I want to solve the following:

$$ x \equiv 5 \text{ (mod 6)}\\ x \equiv 7\text{ (mod 15)} $$ I tried breaking the system as:

$$ x\equiv5\text{ (mod 2)}\\ x\equiv5\text{ (mod 3)}\\ x\equiv7\text{ (mod 3)}\\x\equiv7\text{ (mod 5)} $$ and using the values of $x \text{ (mod 3)}$, I got a contradiction, but clearly $x =22$ is a solution. Can you please help me find where I have gone wrong? Thanks.

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  • $\begingroup$ Your approach is good; see my answer below $\endgroup$ Mar 1 '20 at 13:03
  • $\begingroup$ So, I guess this system does not have any solution in integers. Am I correct? $\endgroup$
    – Kashif
    Mar 1 '20 at 13:20
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    $\begingroup$ Yes, you are correct $\endgroup$ Mar 1 '20 at 13:25
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$x=22$ is not a solution, since $22\not\equiv5\pmod6$.

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