A question about quadratic residues

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The discussion focuses on proving that for an odd integer \( a \), the congruence \( X^2 \equiv a \mod 2 \) is always solvable with exactly one incongruent solution. Furthermore, it establishes that \( X^2 \equiv a \mod 4 \) is solvable if and only if \( a \equiv 1 \mod 4 \), resulting in exactly two incongruent solutions modulo 4. The relationship between these congruences and the general proposition that \( X^2 \equiv a \mod p \) has either no solution or two solutions for an odd prime \( p \) is also discussed, emphasizing the need for manual verification of specific cases.

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yeland404
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I need to prove that a be a odd integer that congruence X^2\equiva 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\equiva mod 4 is solvable iff a\equiv1 mod 4. in this case ,prove that X^2\equiva 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\equiva mod p has either no solution or two solutions, but p there is an odd prime number. HOw to apply to the queations above?
 
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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
 

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