What $x$ satisfies $x^2 \equiv x \pmod{n}$ for arbitrary $n \in \mathbb{N}$?
3563
Find all roots of $\,x^2\!\equiv x\pmod{\!900}$, i.e all idempotents in $\Bbb Z_{900}$
82571
Number of solutions of $x^2=1$ in $\mathbb{Z}/n\mathbb{Z}$
79219
Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
152591
For which prime numbers $p$ does the congruence $x^2 + x + 1 \equiv 0 \pmod{p}$ have solutions?
121717
Prove that $z^2 \equiv ab\pmod p$ is solvable if and only if both or neither of $x^2 \equiv a \pmod p$, $x^2 \equiv b \pmod p$ are solvable.
80843
Find all solutions to $x^9 \equiv 25 \pmod {29}$
32762
Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
62996
A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
41890
How to get all solutions of $x^2 \equiv$ $a$ mod $p$?
4129
Solvability of $a \equiv x^2 \mod b$
24820
Prove that if $n$ is a perfect square, then either $n \equiv 1 \pmod 4$, or $n \equiv 0 \pmod 4$.
84733
Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
66161
How to solve the equation $n^2 \equiv 0 \pmod{584}$?
127901
If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
47060
Split $n$ into nontrivial factors via a nontrivial square-root of $1\!\pmod{\!n}$
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
21874
Solve $x^3+x \equiv 1 \pmod p$