If $\gcd(a,1729) \neq 1$, show that $a^{37 }\equiv a \pmod {1729}$
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
130653
$p^2$ divides $ab$ and $\gcd(a,b)=1$. Show $p^2$ divides $a$ or $p^2$ divides $b$
59006
Proof of $\gcd(n!+1, (n+1)!+1)=1$?
69279
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
17601
How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
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 $a=b$
112031
Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers.
23590
Show that $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n d^{n-1},a-b)$
96535
Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63}$. Why is this assumption necessary?
37808
Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
55441
Check: For the integers $a,b,c$ show that $\gcd(a,bc)=\gcd(a,\gcd(a,b)\cdot c)$
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
66785
Find $\gcd(p^n-1,p^m+1)$.
32108
$ a\in \Bbb Z_n$ is invertible $\,\Rightarrow\gcd (a,n) =1$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
51858
Showing $\gcd(2^m-1,2^n+1)=1$
86413
Show that $\frac{(m+n-1)!}{m!n!}$ is an integer where m and n are positive integers and gcd(m,n)=1