[Math] How to prove by induction that $9^k – 5^k$ is divisible by 4

Question

Recently had this on a discrete math test, which sadly I think I failed. But the question asked:

Prove that $9^k – 5^k$ is divisible by $4$.

Using the only approach I learned in the class, I substituted $n = k$, and tried to prove for $k+1$ like this:

$$9^{k+1} – 5^{k+1},$$

which just factors to $9 \cdot 9^k – 5 \cdot 5^k$.

But I cannot factor out $9^k – 5^k$, so I'm totally stuck.

Answer 1

${\color{White}{\text{Proof without words.}}}$