Using Chinese Remainder Theorem when the moduli are not mutually coprime.

Question

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.

Answer 1

$x=22$ is not a solution, since $22\not\equiv5\pmod6$.