A few points.

1. You can use the "Edit" feature rather than posting multiple times, it will keep the thread cleaner.

2. You made a silly error of not distributing the -4 properly, so you should have -5 where you currently have +3.

3. I think the problem can be solved by considering that when n is odd, 4n+1 is congruent to 5 (mod 8), and when n is even, 4n+1 is congruent to 1 (mod 8). That is, we have by induction hypothesis

\(\displaystyle 5^n-(4n+1)\equiv0\ (\text{mod}\ 8)\)

and so we can find out what \(\displaystyle 5^n\) is (mod 8), by considering the two cases, n odd or even.

4. The way to get exponents with more than one character in LaTeX is like this (hover mouse over it to see the code): \(\displaystyle x^{12345}\).

Edit: I had a typo; the -4 in red above used to say -1.