Divisor of Integer/Examples/8 divides 3^2n + 7
Theorem
Let $n$ be an integer such that $n \ge 1$.
Then:
- $8 \divides 3^{2 n} + 7$
where $\divides$ indicates divisibility .
Proof 1
The proof proceeds by induction.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
- $8 \divides 3^{2 n} + 7$
Basis for the Induction
$\map P 1$ is the case:
\(\ds 3^{2 \times 1} + 7\) | \(=\) | \(\ds 3^2 + 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 + 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 16\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 \times 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 8\) | \(\divides\) | \(\ds 3^{2 \times 1} + 7\) |
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:
- $8 \divides 3^{2 k} + 7$
from which it is to be shown that:
- $8 \divides 3^{2 \paren {k + 1} } + 7$
Induction Step
This is the induction step:
\(\ds 3^{2 \paren {k + 1} } + 7\) | \(=\) | \(\ds 3^{2 k} \times 3^2 + 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3^{2 k} + 7 - 7} \times 3^2 + 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3^{2 k} + 7} \times 3^2 - 7 \times \paren {3^2 - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 r \times 3^2 - 8 \times 7\) | where $8 r = 3^{2 k} + 7$: by the induction hypothesis: $8 \divides 3^{2 k} + 7$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 8 \paren {3^2 r - 7}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 8\) | \(\divides\) | \(\ds 3^{2 \paren {k + 1} } + 7\) | Definition of Divisor of Integer |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\forall n \in \Z_{\ge 1}: 8 \divides 3^{2 n} + 7$
$\blacksquare$
Proof 2
From Integer Less One divides Power Less One, we have that:
- $\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$
Hence we have the special case where $m = 3^2$:
- $8 \divides 3^{2 n} - 1$
from which it follows immediately that:
- $8 \divides 3^{2 n} - 1 + 8 = 3^{2 n} + 7$
$\blacksquare$