find all the solutions: chinese remainder + fermat theorem

Question

I have following congruences, in first congruence I know I have to use the Fermat theorem for finding the solution but I couldn't understand how, because most of the samples in internet using some numbers instead of x. could somebody please explain it for me in simple way?

1) $x^2 \equiv 1 \pmod{3}$

$3x \equiv 6 \pmod{15}$

2) what about the following case? $x^3 \equiv 1 \pmod{3}$

by the way I know this link but as I said it explain by numbers.

Answer 1

For (1) You know by fermat's little theorem that $x$ must be coprime with $3$ and so you have $x=1,2$ and so $x \equiv 1 \pmod 3 \space \textbf{or} \space x \equiv 2 \pmod3$ And you can even try it $1^2 \equiv 1 \pmod3$ $\space$ and $\space $ $2^2 \equiv \pmod 3$

now you have $3x \equiv 6 \pmod{15} $

And since $3,6$ and $15$ are all divisible by $3$ then You can divide the congruence by $3$ to get $x \equiv 2 \pmod5 $ Now how can you use the Chinese remainder theorem ?

Answer 2

the answer is any x =2+5(3k) or x=2+5(1+3k). For the second equation gives x-2=5k or x=2+5k and upon squaring we get k=0 or 1 mod 3. The result follows.