Use induction to show that if $n\geq 0$ then $5^n\equiv 1+4n\pmod{16}$

Question

I was checking the following number theory exercise:

Use induction to show that if $n\geq 0$ then $5^n\equiv 1+4n\pmod{16}$

Applying the induction hypotesis I'm starting from here:

$16\mid 5^{n+1}-5-4n$

I don't know how to proceed from there or to go ahead in the exercise, any help will be really appreciated.

Answer 1

Assume for some $n$, $$5^n\equiv 4n+1\pmod{16}.$$ Then $$5^{n+1}=5(5^n)\equiv5(4n+1)=20n+5\equiv4n+5=4(n+1)+1\pmod{16}.$$

Answer 2

Hint $$5^{n+1}-5-4n- 5 \cdot (5^n-1-4n)=16n$$