Help proving this Proposition

Question

For every natural number $n$, the integer $6^{2n+1}+8^{3n}$ is divisible by 7.

I handled the base case quite well, but got stuck on the induction step. Any help would be greatly appreciated.

Answer 1

Write: $6^{2n+1} + 8^{3n} = (6^{2n+1} + 1^{2n+1}) + (8^{3n} - 1^{3n})$ and the answer follows because:

$6^{2n+1} + 1^{2n+1} = (6+1)(....)$

$8^{3n} - 1^{3n} = (8-1)(....)$.

Answer 2

induction hypothesis: 6^(2m+1)+8^(3m)=7k

inductive step: : 6^(2(m+1)+1) + 8^(3(m+1))

               =(6^(2m+1))*36 + (8^3m)*512

               =(7k-(8^3m))*36 + (7*68*(8^3m))

               =7 (68*(8^3m)+ k)

               =7v