Prove $ax \equiv1 \textrm{ mod } 10^k \implies aN(x) \equiv1 \textrm{ mod } 10^{2k}$
39953
$a\mid b,\ c\mid d\,\Rightarrow\ ac\mid bd $ $\ \, \bf\small [Divisibility\ Product\ Rule]$
17176
Proof that $x^n \mod b = (x \mod b)^n$
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
145722
Prove there is a bijection between the set of $R[x]$-module structures on $M$ and $\mathrm{End}_{R-\mathsf{Mod}}(M)$.
19071
solution set for congruence $x^2 \equiv 1 \mod m$
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
42780
Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.
139503
How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
100134
Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC
39082
Proving $x^\alpha-\alpha x \le 1- \alpha $
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
110703
How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m?
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
84021
How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?
94855
Prove if a (mod n) has a multiplicative inverse, then it's unique
62996
A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.