Prove That $3^n + 8^n$ is Not Divisible by $5$ (Using Induction) Prove that $3^n+8^n$ is not divisible by 5.
I know that this can be proved by using congruence and I am providing the proof by congruence below. But is there any way to Prove It By Induction.
The proof by congruence goes like this:
$3\equiv 3\pmod 5 \\ 3^2 \equiv 4\pmod 5 \\ 3^3\equiv 7\pmod 5 \\ 3^4\equiv 1\pmod 5 \\ 3^5\equiv 3\pmod 5$
Also,
$8\equiv 3\pmod 5 \\ 8^2 \equiv 4\pmod 5 \\ 8^3\equiv 7\pmod 5 \\ 8^4\equiv 1\pmod 5 \\ 8^5\equiv 3\pmod 5$
Adding the congruence up (since the same cycle repeats after the 4th power) none of them are divisible by 5 or equal to 0.
But I need a proof by Induction.
Any help will be appreciated.

You can go like this. For the base case, let $n=1$. Then
$$3^n+8^n = 3+8 = 11,$$
which is not divisible by $5$.
For the induction step, let $n\geq 1$ be arbitrary and assume that $3^n+8^n$ is not divisible by $5$. Now
\begin{align*}
3^{n+1} + 8^{n+1} &= 3\cdot 3^n + 8\cdot 8^n\\
&= 3\cdot 3^n + (3+5)\cdot 8^n\\
&= 3\cdot 3^n + 3\cdot 8^n + 5\cdot 8^n\\
&= 3(3^n + 8^n) + 5\cdot 8^n.
\end{align*}
The first term is not divisible by $5$ by the induction hypothesis, and because $3$ and $5$ are relatively prime, so since $5$ divides the second term, it does not divide their sum.
