1. **State the problem:** We need to prove that for any positive integer $n$, the expression $7^n - 1$ is divisible by 6. 2. **Rewrite the problem:** To say $7^n - 1$ is divisible by 6 means $6 \mid (7^n - 1)$, or equivalently, $7^n - 1 \equiv 0 \pmod{6}$. 3. **Analyze modulo 6:** Since $7 \equiv 1 \pmod{6}$ (because $7 - 6 = 1$), we can replace $7$ by $1$ modulo 6. 4. **Apply exponentiation:** Then $7^n \equiv 1^n \equiv 1 \pmod{6}$. 5. **Subtract 1:** Therefore, $7^n - 1 \equiv 1 - 1 \equiv 0 \pmod{6}$. 6. **Conclusion:** This shows $7^n - 1$ is divisible by 6 for all positive integers $n$. **Final answer:** $7^n - 1$ is divisible by 6 for every positive integer $n$.