Prove that $a^2 \equiv 4 \mod 3^m$ if and only if $a \equiv 2 \mod 3^m$ or $a \equiv −2 \mod 3^m$
86594
If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$
20161
Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.
11693
If $p \mid m^p+n^p$ prove $p^2 \mid m^p+n^p$
88036
Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$
102234
Show $a^p \equiv a \mod 2p$
27063
Prove that if n is an odd positive integer , then $n^2\equiv1\mod8$
15191
If $a^2$ divides $b^2$, then $a$ divides $b$
114346
Why is $a^{-1}$ mod $p$ equal to $a^{p-2}$ mod $p$?
103499
$x^2 \equiv a \mod p$, $\ x^2 \equiv b \mod p$, and $x^2 \equiv ab \mod p$, prove either all three are solvable or exactly one
44247
For $a^3+b^3+c^3=3$ prove that $a^4b+b^4c+c^4a\leq3$
136367
Prove $ ∀a, b, c ∈ Z,$ if $ab + ac ≡ 3$ (mod 6) then b $\not\equiv$ c (mod 6).
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
77064
Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
101057
Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$
146851
Prove that $-3$ is a quadratic residue mod an odd prime $>3$ if and only if $p$ is of the form of $6n+1$
11989
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
40968
show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$