$ a \equiv b \ (\text{mod} \ p) \Rightarrow a^{n} \equiv b^{n} \ (\text{mod} \ p)?$
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
4972
Mod of numbers with large exponents [modular order reduction]
43246
Showing that if $a<b$, then $a^{1/n} < b^{1/n}$
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
70861
$\mathbb P^n\times \mathbb P^m\not\cong \mathbb P^{m+n}$
58803
${n \choose k} \bmod m$ using Chinese remainder theorem?
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
94855
Prove if a (mod n) has a multiplicative inverse, then it's unique
83542
A natural problem in $\textrm{S}_2^\textrm{P}$?
23153
$\bmod p\!:\ x^2\equiv 1\iff x\equiv \pm1\ $ for prime $p$
122741
$x^2$ is congruent to $-1 \bmod p$
4129
Solvability of $a \equiv x^2 \mod b$
13317
show that $a^{12} \equiv 1 \mod 32760$ for each $a \in \mathbb{Z}$ such that $\text{gcd}(a, 32760) = 1$
13904
$p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?
91935
If $x \equiv a \pmod {p_1}$ and $x\equiv a \pmod{p_2}$, then is it true that $x\equiv a \pmod{p_1p_2} ?$