Having a bit of trouble with this one. At the point markedA, I'm not sure I'm going in the right direction. Immediately after that line I haveB, which I'm showing as a possible alternative, but I'm unsure of that approach as well. Can anyone help me work out which, if either, of the two methods is correct?

Many thanks.

Q.$\displaystyle 9^n-5^n$ is divisible by 4, for $\displaystyle n\in\mathbb{N}_0$

Attempt:Step 1:For $\displaystyle n=1$...

$\displaystyle 9^1-5^1=4$, which can be divided by $\displaystyle 4$.

Therefore, $\displaystyle n=1$ is true...

Step 2:For $\displaystyle n=k$...

Assume $\displaystyle 9^k-5^k=4\mathbb{Z}$, where $\displaystyle \mathbb{Z}$ is an integer...1

Show that $\displaystyle n=k+1$ is true...

i.e. $\displaystyle 9^{k+1}-5^{k+1}$ can be divided by $\displaystyle 4$

$\displaystyle 9^{k+1}-5^{k+1}$ => $\displaystyle 9^1\cdot9^k-5^1\cdot5^k$ =>

A$\displaystyle (9-5)[9^k-5^k]$ => $\displaystyle 4[4\mathbb{Z}]$...from1above => $\displaystyle 4[4\mathbb{Z}]$, which can be divided by 4

B$\displaystyle 9[4\mathbb{Z}-5^k]-5[9^k-4\mathbb{Z}]$

Thus, assuming $\displaystyle n=k$, we can say $\displaystyle n=k+1$ is true & true for $\displaystyle n=2,3,$... & all $\displaystyle n\in\mathbb{N}_0$