[Math] 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

You could start by rewriting the congruence in equation form as follows

$$ 4x\equiv 2 \pmod 6 \iff 4x=2+6\cdot k $$

now you are dealing with a usual equation, here you can divide by $2$ to get

$$ 2x=1+3\cdot k \iff 2x\equiv 1\pmod 3\iff \ 2\cdot2x\equiv 2\cdot 1\pmod 3 $$

which means

$$ x\equiv 2\pmod 3. $$

But remember we needed a solution modulo $6$, so we need to check which numbers in the set $\{0,1,2,3,4,5 \}$ satisfies the above congruence an we easily find that $2$ and $5$ is congruent to $2$ modulo $3$ and these are our solutions.