We are tasked with proving by mathematical induction that $8^n - (-5)^n$ is divisible by 13 for all positive integers $n$.

Step 1: Base Case $n = 1$

For $n = 1$: $8^1 - (-5)^1 = 8 - (-5) = 8 + 5 = 13$ Clearly, 13 is divisible by 13. Thus, the base case holds.

Step 2: Inductive Hypothesis

Assume that for some positive integer $k$, the statement holds. That is, assume: $8^k - (-5)^k \text{ is divisible by 13}$ This means there exists an integer $m$ such that: $8^k - (-5)^k = 13m$

Step 3: Inductive Step

We need to prove that $8^{k+1} - (-5)^{k+1}$ is divisible by 13.

We start with: $8^{k+1} - (-5)^{k+1} = 8 \cdot 8^k - (-5) \cdot (-5)^k = 8 \cdot 8^k - 5 \cdot (-5)^k$

Now, use the inductive hypothesis: $8^k - (-5)^k = 13m$ We rewrite $8^{k+1} - (-5)^{k+1}$ as: $8 \cdot 8^k - 5 \cdot (-5)^k = 8(8^k - (-5)^k) + (8 \cdot (-5)^k - 5 \cdot (-5)^k)$ Using the inductive hypothesis $8^k - (-5)^k = 13m$, we get: $8 \cdot 8^k - 5 \cdot (-5)^k = 8 \cdot 13m + (8 - 5)(-5)^k = 104m + 3(-5)^k$ Since $(-5)^k$ is either $5^k$ or $-5^k$, it follows that $3(-5)^k$ is divisible by 13 for all $k$. Thus, the entire expression is divisible by 13.

Therefore, by the principle of mathematical induction, $8^n - (-5)^n$ is divisible by 13 for all positive integers $n$.

Would you like further details or have any other questions?

Here are 5 related questions for further practice:

1. How would you prove the same statement for a different modulus, say 7?
2. Can you generalize this proof for any expression of the form $a^n - b^n$ being divisible by a prime?
3. How does the divisibility property change if we use negative exponents?
4. Can this proof be done using other proof techniques aside from induction?
5. How does modular arithmetic simplify such problems?

Tip: When proving a divisibility statement using induction, always check how the inductive step builds from the hypothesis carefully, especially in cases involving powers and modular arithmetic.