# Divisor of Integer/Examples/80 divides 9^2n - 1/Proof 1

Jump to navigation Jump to search

## Theorem

Let $n \in \Z_{\ge 0}$ be a non-negative integer.

Then:

$80 \divides 9^{2 n} - 1$

where $\divides$ denotes divisibility.

## Proof

The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the equivalent proposition:

$80 \divides 9^{2 n} - 1$

where $\divides$ denotes divisbility

$\map P 0$ is the case:

 $\ds 9^{2 \times 0} - 1$ $=$ $\ds 9^0 - 1$ $\ds$ $=$ $\ds 1 - 1$ Zeroth Power of Real Number equals One $\ds$ $=$ $\ds 0$ $\ds \leadsto \ \$ $\ds 80$ $\divides$ $\ds 9^{2 \times 0} - 1$ Integer Divides Zero

Thus $\map P 0$ is seen to hold.

### Basis for the Induction

$\map P 1$ is the case:

 $\ds 9^{2 \times 1} - 1$ $=$ $\ds 9^2 - 1$ $\ds$ $=$ $\ds 81 - 1$ $\ds$ $=$ $\ds 80$ $\ds \leadsto \ \$ $\ds 80$ $\divides$ $\ds 9^{2 \times 1} - 1$ Integer Divides Itself

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:

$80 \divides 9^{2 k} - 1$

from which it is to be shown that:

$80 \divides 9^{2 \paren {k + 1} } - 1$

### Induction Step

This is the induction step:

From the induction hypothesis we have that:

$80 \divides 9^{2 k} - 1$

Hence by definition of divisibility, we have:

$\exists r \in \Z: 9^{2 k} - 1 = 80 r$

and so:

$(1): \quad \exists r \in \Z: 9^{2 k} = 80 r + 1$

 $\ds 9^{2 \paren {k + 1} } - 1$ $=$ $\ds 9^2 \times 9^{2 k} - 1$ $\ds \leadsto \ \$ $\ds \exists r \in \Z: \,$ $\ds 9^{2 \paren {k + 1} } - 1$ $=$ $\ds 81 \times \paren {80 r + 1} - 1$ from $(1)$ $\ds$ $=$ $\ds 81 \times 80 r + 80$ algebra $\ds$ $=$ $\ds 80 \paren {81 r + 1}$ algebra

Hence by definition of divisibility:

$80 \divides 9^{2 \paren {k + 1} } - 1$

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:

$\forall n \in \Z_{\ge 0}: 80 \divides 9^{2 n} - 1$

$\blacksquare$