Five to the n minus one is divisible by four

Statement

Prove that $5^n - 1$ is divisible by $4$ for every integer $n \ge 1$.

Solution

By induction. Base ($n=1$): $5^1 - 1 = 4$, divisible by $4$.

Step: assume $4 \mid (5^k - 1)$, i.e. $5^k - 1 = 4t$ for some integer $t$. Then $5^{k+1} - 1 = 5 \cdot 5^k - 1 = 5\,(5^k - 1) + 4 = 5 \cdot 4t + 4 = 4\,(5t + 1).$ So $4 \mid (5^{k+1}-1)$. By the principle of induction the claim holds for all $n \ge 1$. $\blacksquare$

Alternatively: $5 \equiv 1 \pmod 4$, so $5^n \equiv 1^n = 1 \pmod 4$, hence $5^n - 1 \equiv 0 \pmod 4$.