How to solve system of congruence with common divisor? How to solve these (4) systems of congruences? $$ \begin{cases} x=3,5\pmod{8}\\\ x=5,7\pmod{12} \end{cases} $$ I was thinking about using CRT but $\gcd\left(8,12\right)>1$. What can I do? **Clarification:** This notation represents 4 different systems. One of them for example is $$ \begin{cases} x=3\pmod{8}\\\ x=7\pmod{12} \end{cases} $$

As chí trunch châo mentioned. If you work $\textrm{mod}\ 24$, you should see that being $3\ \textrm{mod}\ 8$ gives $(3,11,19)\ \textrm{mod}\ 24$ (excuse my notation) and $5\ \textrm{mod}\ 8$ gives $(5,13,21)\ \textrm{mod}\ 24$. Moreover, $5\ \textrm{mod}\ 12$ gives $(5,17)\ \textrm{mod}\ 24$ and $7\ \textrm{mod}\ 12$ gives $(7,19)\ \textrm{mod}\ 24$. Looking at these possibilities and its intersection, it is not hard to see that the only two possibilities remaining are $5\ \textrm{mod}\ 24$ and $19\ \textrm{mod}\ 24$ which are solutions to the combinations $(5,5)$ and $(3,7)$, respectively.