<s>[INST] Proving that $5^n - 1$ is divisible by $4$ by mathematical induction. I have done it, but I am not sure that the inductive step is right. Can anybody please clear me about it? 
Basic steps as:
Taking $n=1$: $p(1)=5-1=4$.
Inductive hypothesis:  Assume the statement is true for $p(k)$.  $5^k - 1$ is divisible by $4$.
Inductive steps: We must show $p(k+1)$ is true when $p(k)$ is true.
\begin{align*}
& 5^k -1 + 5^{k+1} -1\\
& 5^k -1 + 5.5^{k} -1\\
& (5^k -1) + 4
\end{align*}
 [/INST] The idea is $$5^{k+1} - 1 = 5\cdot 5^{k} - 1 = 4\cdot(5^k) + (5^k - 1)$$ 
We have $4\cdot(5^k)$ is obviously divisible by $4$ and $5^k - 1$ is divisible by $4$ by inductive hypothesis 
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