Simple Proof by Induction

**demode** · March 2nd 2011, 09:21 PM

Using mathematical induction I want to show that $3^{6n}-2^{6n}$ is divisible by $35$ , $\forall n \in \mathbb{N}$ .

The base case n=1 is true: $3^6-2^6=665$ , 665/35=19.

Inductive step: Suppose $3^{6k}-2^{6k}$ is divisible by 35 for some $k \in \mathbb{N}$ . That means $3^{6k}-2^{6k} = 35m$ for some m. Then

$3^{6(k+1)}-2^{6(k+1)}= 3^{6k}.3^6-2^{6k}.2^6$

Now how can I simplify this to factor out $3^{6k}-2^{6k}$ to show that it's divisible by 35 for k+1?