Prove that $\gcd(a^{2^m}+1,a^{2^n}+1)=\begin{cases} 1,& \text{ if } a\ \text{is even;} \\ 2 & \text{ if } a\ \text{is odd.} \end{cases}$.
34059
$(a^{2^n}+1,a^{2^m}+1)=1 or 2$
128199
Proving gcd(a, a+2) is 1 or 2 for every integer a?
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
59006
Proof of $\gcd(n!+1, (n+1)!+1)=1$?
126358
Prove for $\gcd$ of two integers $a,b$, that $(a^n,b^n) = (a,b)^n$
97469
How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
18772
If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.
135779
Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
72424
Suppose 1 ≤ k ≤ n − 1 and gcd(k, n) = 1. Prove that gcd(n − k, n) = 1
82777
If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$?
31895
Prove that if $a$ and $b$ are odd, coprime numbers, then $\gcd(2^a +1, 2^b +1) = 3$
107095
Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$
147246
If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
14693
$d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule]
73391
Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$
21788
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$