Modular division

Modular division by $b$ modulo $m$ means multiplication by a modular inverse $b^{-1}$. This is defined uniquely for every numerator exactly when $\gcd (b,m)=1$. For example, $3^{-1}\equiv 5(\mathrm{mod}7)$, so $2/3\equiv 2\cdot 5\equiv 3(\mathrm{mod}7)$.

If $b$ is not invertible, solve the linear congruence $bx\equiv a(\mathrm{mod}m)$ directly. It has solutions exactly when $d=\gcd (b,m)$ divides $a$, and then has $d$ solutions modulo $m$. Thus $2x\equiv 2(\mathrm{mod}6)$ gives $x\equiv 1,4(\mathrm{mod}6)$; division by two cannot select a unique residue. Cancellation instead reduces the modulus to $m/d$.