Solution verification: Find all possible $x$ that satisfy $5x\equiv 1\;\;(mod\;\;6)$.

Question

Can someone please verify/explain why this solution works? I just know how to do it but I want to know why it works!

$5x\equiv 1\;\;(mod\;\;6)$

$5x = 1 + 6y$

$5n = -4 + y \;\;(mod \;\;5)$

$y = 5n + 4$

$5x = 1 + 6(5n + 4) = 1 + 5(6n) + 24 = 25 + 5(6n)$

$x = 5 + 6n, \;\; n\in \mathbb{Z}$

Answer 1

Solving $$5x\equiv 1\;\;(mod\;\;6)$$ means finding all $x\in (0,1,2,3,4,5)$ which satisfy $ 5x=6k+1.$

Upon checking those numbers, we find that $x=5$ is the only one satisfying the equation.

Therefore the answer is $x=6k+5.$

Answer 2

$5x \equiv 1 \pmod 6$

So

$5x \equiv 1 \pmod 2$ AND $5x \equiv 1 \pmod 3$

So

$x \equiv 1 \pmod 2$ AND $x \equiv -1 \pmod 3$

So

$x \equiv -1 \pmod 6 \equiv 5 \pmod 6$

Answer 3

Alternative hint:

$$\,5x\color{red}{+x}\equiv 1\color{red}{+x} \iff 6\,x \equiv 1+x \iff 0 \equiv 1+x \iff x \equiv -1 \iff x \equiv 5 \pmod{6}\,$$