I'm looking for a power of $2$ that ends in $0$, so $2^n = k\cdot 10$, where $n \in \mathbb{Z}$ and $k \in \mathbb{Z}$. It doesn't seem to exist. As an equation with two variables, I was looking for a second equation, but just can't think of any.

I also trialled a few consecutive powers of $2$ and noticed that there seems to be a sequence perpetually ending in $2, 4, 8, 6, 2, 4, 8, 6, \cdots$

How can I prove mathematically that there will never be a power of $2$ that ends in $0$? Is there a simply way to explain that?