If $gcd(n,m)=1$ and $x\equiv y(mod\phi(m))$ then $n^{x}\equiv n^{y}(modm)$.
4972
Mod of numbers with large exponents [modular order reduction]
35945
Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
96535
Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63}$. Why is this assumption necessary?
32463
$ax\equiv b\pmod{\! n}$ has a unique solution if $\gcd(a,n) = 1$
37346
If $m, n$ be positive integers, prove that $\phi(mn)=\phi((m,n))\phi([m,n])$, where $(m,n)=$ gcd of $m, n$ and $[m, n]=$ lcm of $m, n$.
157711
Number Theory: Prove that $m | \phi(a^m - 1)$
62996
A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
65218
Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$
66785
Find $\gcd(p^n-1,p^m+1)$.
59566
Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
14693
$d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule]
61825
Congruence class $[a]$ modulo $m$, $\gcd(x, m) = \gcd(a, m)$
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
126280
In a P.I.D., if $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$ with $\gcd(m,n) = 1$, then $a=b$
17176
Proof that $x^n \mod b = (x \mod b)^n$
138058
Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.