If $a \equiv r_i^2 \mod p_i$ for every prime divisor $p_i$ of $b$, then $a$ is a square modulo $b$.
4129
Solvability of $a \equiv x^2 \mod b$
84733
Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
94917
Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
13478
Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
64453
Prove if $(a,p)=1$, then $\{a,2a,3a,...,pa\}$ is a complete residue system modulo $p$.
6160
How to prove a number $a$ is quadratic residue modulo $n$?
79219
Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
94017
If p is a prime and a ∈Z. How do we prove that the congruence x^x ≡ a (mod p) is solvable?
36043
If $ab \equiv r \pmod{p}$, and $x^2 \equiv a \pmod{p}$ then $y^2 \equiv b \pmod{p}$ for which condition of $r$?
131971
If $k>0$ is a positive integer and $p$ is any prime, when is $\mathbb Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in\mathbb Z_p\}$ a field.
32762
Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
65581
$m$ is a perfect square iff $m$ has an odd number of divisors?
27063
Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
147398
If every element of $H$ and $G/H$ is a square, then prove that so is every element of $G$.
50596
$p_1p_2...p_n +1$ is not square for $p_k = k$'th prime
24820
Prove that if $n$ is a perfect square, then either $n \equiv 1 \pmod 4$, or $n \equiv 0 \pmod 4$.