Congruence $4x \equiv 2 \pmod 6$

Question

Is there a solution for the congruence $4x \equiv 2 \pmod 6$ ? And how can I find inverse element for $4$, when I can not use Extended Euclidean algorithm, because $6$ and $4$ are divisible by $2$.

Thanks for the answers!

Answer 1

$ 4x \equiv 2 \bmod 6 $ implies

• $ 4x \equiv 2 \bmod 2 $, which does not give any information

• $ 4x \equiv 2 \bmod 3 $, which reduces to $ x \equiv 2 \bmod 3 $

Conversely, every number of the form $x=3k+2$ is a solution of $ 4x \equiv 2 \bmod 6 $.

Therefore, $ 4x \equiv 2 \bmod 6 $ iff $x \equiv 2,5 \bmod 6 $.