Proving That (9^n - 1) is Divisible by 8 Using Mathematical Induction

Mathematical induction is a powerful tool in number theory and discrete mathematics for proving statements that apply to all positive integers. One classic problem is proving that (9^n - 1) is divisible by 8 for any positive integer (n). This article explores two different methods to solve this problem: the first uses direct mathematical induction, while the second applies a different approach through algebraic manipulation.

Proof Using Direct Induction

Let's dive into the direct proof using mathematical induction.

Base Case

For (n 1), we have:

$$9^1 - 1 9 - 1 8$$

The statement is true for (n 1) because 8 is divisible by 8.

Inductive Step

Assume the statement is true for some positive integer (n k). That is:

$$9^k - 1 8m$$ for some positive integer (m).

We now need to show that the statement is true for (n k 1). That is, we need to prove:

$$9^{k 1} - 1 8p$$ for some positive integer (p).

Starting from the left-hand side (LHS), we have:

(1) $$9^{k 1} - 1 9 cdot 9^k - 1$$

(2) Using the inductive hypothesis, we get:

(3) $$9 cdot 9^k - 1 9 cdot (8m 1) - 1 9 cdot 8m 9 - 1 9 cdot 8m 8 8(9m 1)$$

(4) Hence, (9^{k 1} - 1) is divisible by 8, and we can write:

(5) $$9^{k 1} - 1 8p$$ for some positive integer (p).

This completes the inductive step, proving the statement for all positive integers (n).

Alternative Proof Using Algebraic Manipulation

An alternative method involves algebraic manipulation to show that (9^n - 1) is divisible by 8 for any positive integer (n).

Base Case

For (n 0), we have:

$$9^0 - 1 1 - 1 0$$

The statement is true for (n 0) because 0 is divisible by 8.

Inductive Step

Assume the statement is true for some nonnegative integer (n k). That is:

$$9^k - 1 8m$$ for some nonnegative integer (m).

We now need to show that the statement is true for (n k 1). That is, we need to prove:

$$9^{k 1} - 1 8p$$ for some nonnegative integer (p).

Starting from the left-hand side (LHS), we have:

(1) $$9^{k 1} - 1 9 cdot 9^k - 1$$

(2) Using the inductive hypothesis, we get:

(3) $$9 cdot 9^k - 1 9 cdot (8m 1) - 1 72m 9 - 1 72m 8 8(9m 1)$$

(4) Hence, (9^{k 1} - 1) is divisible by 8, and we can write:

(5) $$9^{k 1} - 1 8p$$ for some nonnegative integer (p).

This completes the inductive step, proving the statement for all nonnegative integers (n).

Conclusion

The two methods presented here both demonstrate that the expression (9^n - 1) is always divisible by 8 for all positive integers (n). This proof not only strengthens understanding of the concept of mathematical induction but also highlights different ways to achieve the same conclusion.

Additional Resources

For further exploration of induction proofs and divisibility, consider revisiting the Wikipedia page on mathematical induction and related topics on Brilliant.