Answer
See explanations.
Work Step by Step
Step 1. Prove the statement is true when $n=1$: $8^1-3^1=5$ is divisible by 5, thus it is true for $n=1$.
Step 2. Assume the statement is true when $n=k$: we have $8^k-3^k$ is divisible by 5. We can assume $8^k-3^k=5m$ or $8^k=5m+3^k$ where $m$ is an integer.
Step 3. Prove it is true for $n=k+1$: $8^{k+1}-3^{k+1}=8(5m+3^k)-3^{k+1}=40m+(8-3)3^k=40m+5\times3^k$. As both $40m$ and $5\times3^k$ are divisible by 5, their sum is also divisible by 5.
Thus, the statement is also true for $n=k+1$
Step 4. Conclusion: with mathematical induction, we have proved that the statement is true for all natural numbers $n$.
