elementary number theory - Proof of divisibility using modular arithmetic: $5mid 6^n - 5n + 4$

Prove that:

$$6^n - 5n + 4 \space \text{is divisible by 5 for} \space n\ge1$$

Using Modular arithmetic. Please do not refer to other SE questions, there was one already posted but it was using induction, I want to use this number theory method.

Obviously we have to take $\pmod 5$

So:

$$6^n - 5n + 4 \equiv x \pmod 5$$

All we need to do prove is prove $x = 0$

How do we do that? I just need a hint, I am not sure how to solve congruences. Some ideas will be helpful.

Thanks!

Answer

Hint:-
$6\equiv1 \pmod 5\implies 6^n\equiv1\pmod 5\tag{1}$

$-5(n-1)\equiv 0\pmod 5\tag{2}$

Solution:-

$(1)+(2)$ gives,

$$6^n-5n+4\equiv0\pmod 5$$