If $a_0,....a_n$ are the set of residues modulo $n$ and $\gcd(a,n)=1$, prove that $aa_0,aa_1,....,aa_n$ is also a complete set of residue modulo $n$.
105225
Number Theory: Complete set of residues modulo $n$
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
119051
Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $\text{gcd}(m,n) = 1$, $m$ is prime.
38404
Prove $\gcd(a+b, a-b) = 1$ or $2\,$ if $\,\gcd(a,b) = 1$
6012
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
155784
If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers
96326
Number of $q$-th residues modulo $n$
154959
Proof of associativity of GCD?
76680
Prove that if $\gcd(a,c)=1$ and $\gcd(b,c)=1$ then $\gcd(ab,c)=1$
96095
If $\{a_1,\dots,a_{\phi(n)}\}$ is a reduced residue system, what is $a_1\cdots a_{\phi(n)}$ congruent to?
7151
Let $a,m,n \in \mathbf{N}$. Show that if $\gcd(m,n)=1$, then $\gcd(a,mn)=\gcd(a,m)\cdot\gcd(a,n)$.
113641
Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)
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 $
63769
Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$)
64742
Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
8160
What is $\gcd(0,a)$, where $a$ is a positive integer?