Induction Proof

**DiscreteW** · February 26th 2009, 05:26 PM

Prove that for all $n \geq 1$ , prove $5$ divides $8^n - 3^n$ using inducation.

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So I know the inductive step:

Assume $5$ divides $8^k-3^k$

Show $5$ divides $8^{k+1} - 3^{k+1}$

So, $8^k - 3^k = 5p$ , for some integer $p$ . This is the inductive hypothesis.

$8^{k+1}-3^{k+1} = \left(8^k\cdot 8 - 3^k\cdot 3\right) = \left(8^k\cdot (5+3) - 3^k\cdot 3\right) = \left(5\cdot 8^k + 3\cdot 8^k\right) - 3^k\cdot 3$

I'm stuck from here. Thanks for the help.