<s>[INST] Prove by induction $6\cdot 7^n -2\cdot 3^n$ is divisible by $4$ for $n\ge 1$ Prove by induction $6\cdot 7^n -2\cdot 3^n$ is divisible by $4$ for $n\ge 1$.
I am struggling on this problem very much.
So far I have Basis case = $6\cdot 7^1 - 2\cdot3^1 = 36$ which is divisible by $4$
Assume for a $n$ that $6\cdot 7^n-2\cdot3^n$ is divisible by $4$.
After that I am not sure what to do.
 [/INST] $$6\cdot7^{n+1}-2\cdot3^{n+1}=7(6\cdot7^n-2\cdot3^n)+14 \cdot 3^n-2 \cdot 3^{n+1}=7(6\cdot7^n-2\cdot3^n)+8 \cdot 3^n$$
The first term is divisible by 4  by  inductive assumption and $8= 4 \cdot 2$
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