$8^n-3^n$ Divisible by 5 - Proof Verification. Statement: $\frac{8^k-3^k}{5}=M, M\in\mathbb{N}$
Base case: $P(1): \frac{8-3}{5}=1\in\mathbb{N}$
Assume $P(n): \frac{8^n-3^n}{5}=N$
Then, $P(n+1)=8^{n+1}-3^{n+1}=5K$, where $K$ is in terms of $M$
Writing LHS in terms of $N$: 
$8^n-3^n=5N \to 8\cdot8^n-8*3^n=40N$ 
$8^{n+1}-3^{n+1}=40M+5\cdot3^{n+1}$
Dividing through by $5$:
$\frac{8^{n+1}-3^{n+1}}{5}=8M+3^{n+1}=K$
, where $K$ is clearly an integer, as $M$ and $N$ Are defined to be integers.
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Questions:


*

*Is this proof valid?

*Adding to that, is that last deduction of $K\in\mathbb{N}$ fair?


Hint: Use that if $$8\equiv 3\mod 5$$ then $$8^n\equiv 3^n\mod 5$$
