1. **Problem:** Prove by mathematical induction that $6^n - 1$ is divisible by 5 for $n \geq 0$. 2. **Base Case:** For $n=0$, calculate $6^0 - 1 = 1 - 1 = 0$, which is divisible by 5. 3. **Inductive Hypothesis:** Assume for some $k \geq 0$, $6^k - 1$ is divisible by 5. That is, $6^k - 1 = 5m$ for some integer $m$. 4. **Inductive Step:** Show $6^{k+1} - 1$ is divisible by 5. $$6^{k+1} - 1 = 6 \cdot 6^k - 1 = 6(6^k) - 1$$ Using the hypothesis: $$6^{k+1} - 1 = 6(5m + 1) - 1 = 30m + 6 - 1 = 30m + 5 = 5(6m + 1)$$ Since $6m + 1$ is an integer, $6^{k+1} - 1$ is divisible by 5. 5. **Conclusion:** By induction, $6^n - 1$ is divisible by 5 for all $n \geq 0$. Final answer: **$6^n - 1$ is divisible by 5 for all $n \geq 0$.**