$a \in \mathbb{N}$ and $p \in \mathbb{P}$ with $gcd(a, p-1)=1$, then for every integer $b$, then $x^a \equiv b$ mod $p$ has a solution
58368
Proof of Unique Solution for Modular Exponentiation In Cryptography
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
38404
Prove $\gcd(a+b, a-b) = 1$ or $2\,$ if $\,\gcd(a,b) = 1$
144851
For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
17176
Proof that $x^n \mod b = (x \mod b)^n$
82938
$\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$.
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
18693
If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
15168
If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
128107
find all $ a \in \mathbb N $ such that $2^a − 1 \equiv 0 \pmod 3$ and $ 4x^2 + 1 \equiv 0 \pmod {\frac{2^a -1}{3}}$ has a solution
36
Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$
151985
If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
5708
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
113641
Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)