I have $ \mathsf x^2 \equiv 1 mod 15 $ They first i thought was, why don't we check which x are congruent to 1? So i replace x with numbers from 0 to 14; after that, i see that
$ \mathsf 1^2 \equiv 1 (15) $
$ \mathsf 4^2 \equiv 1 (15) $
$ \mathsf 11^2 \equiv 1 (15) $
$ \mathsf 14^2 \equiv 1 (15) $
But then i don't know how to proceed. I need to have 4 cases where:
$ \mathsf x \equiv 1 (5) $
$ \mathsf x \equiv 1 (3) $
(in sistem)
$ \mathsf x \equiv 1 (5) $
$ \mathsf x \equiv -1 (3) $
(in sistem)
$ \mathsf x \equiv -1 (5) $
$ \mathsf x \equiv 1 (3) $
(in system)
$ \mathsf x \equiv -1 (5) $
$ \mathsf x \equiv -1 (3) $
in sistem
I'd really like an overall explanation of the whole procedure tho, would be helpful!