A question about quadratic residues

[yeland404]

I need to prove that a be a odd integer that congruence X^2$\equiv$a mod 2
is always solvable with exactly one incongruent solution modulo 2.
this question is linked with (b) let a be an odd integer. Prove that the congruence X^2$\equiv$a mod 4 is solvable iff a$\equiv$1 mod 4. in this case ,prove that X^2$\equiv$a mod 4solutions has exactly two incongruent
solutions modulo 4.

these two seem to link with each other. And the proposition I learn is X^2$\equiv$a mod p has either no solution or two solutions, but p there is an odd prime number. HOw to apply to the queations above?

 

[Office_Shredder]

You should be able to just check these by hand. For example does x^2=2 (mod 4) have any solutions? Just plug in 0,1,2,3 for x and see what you get