How else could we solve the congruence?

Question

How could I solve the congruence $6x \equiv 1 \pmod { 5^4}$?

I wanted to use the formula $x_n=\frac{5^4+1}{6}$, but calculating this number, we see that it is not integer.

How else could we solve the congruence?

Answer 1

$5^4=625\equiv1\pmod6$, so $5\cdot5^4\equiv5\pmod6$, and $5^5\equiv5\pmod6$, i.e., $6\mid 5^5+1$.

Now $5^5+1\equiv1\pmod{5^4}$, so you could just as well solve $6x\equiv5^5+1\pmod{5^4}$.

Answer 2

You could note that $5^4-1=624$ is a multiple of $6$, in fact it is $6\cdot104$. This means that $$6\cdot104\equiv -1 \pmod{5^4}$$ or equivalently $$6^{-1} \equiv -104 \pmod{5^4}$$

Multiplying both sides (of the original equation) by $-104$ then yields that

$$x\equiv -104 \pmod{5^4}.$$