jmc

Subtract 1 from both sides to obtain $8x\equiv 4(\mathrm{mod}12)$. Add 12 to the right-hand side to get $8x\equiv 16(\mathrm{mod}12)$. Now divide both sides by 8, remembering to divide 12 by the greatest common factor of 12 and 8, thus obtaining $x\equiv 2(\mathrm{mod}3)$. This gives $a+m=2+3=\boxed{5}$. Here we have used the fact that $ad\equiv bd(\mathrm{mod}m)$ if and only if $a\equiv b(\mathrm{mod}m/\text{gcd}(m,d))$, for integers $m\ge 2$ and $a$, $b$, and $d$.