Prove for every $k\geq1$ there exist $n$ that satisfies $2^n = -2^k$ (mod $10^k$).

Question

Prove for every $k\geq1$ there exist $n$ that satisfaest $2^n = -2^k$ (mod $10^k$).

By simply trying I saw this is correct but I have no idea how to prove it.

Can someone help?