Solution for $2^n = 12k$

Question

Let $k, n \in \mathbb{Z}$.

How can I find a solution for this problem?

Also, where are some resources to solve a similar problem? I came across this question when attempting to find a solution to $2^n \equiv 0 \mod 12$.

Answer 1

Hint: Do you know many powers of $2$ which are multiples of $3$?

Answer 2

If $$2^n=12k$$ then $$2^{n-2}=3k$$ $$2^{n-2}\equiv 0 \mod{3}$$ $$(-1)^{n}\equiv 0 \mod{3}$$ $$0\in\{-1,1\}$$ Which is not true - contradicts the original equation.