Given $a=qb+r$, prove $\gcd(a,b)=\gcd(b,r)$
190
Why $\gcd(qb+r,b)=\gcd(b,r)$?
4486
If $ar + bs =1$, then $\gcd(a,s) = \gcd(r,b) = \gcd(r,s) = 1$
69279
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
73391
Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$
151434
If integers $a$ and $b$ are not both zero, then $\gcd(a,b)=\gcd(a-b,b)$
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
77278
If $\,\gcd(a,b) = {\rm lcm}(a,b)\,$ then prove $a=b$
31465
Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
97469
How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
113641
Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)
76507
$\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
136064
How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
128820
Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
110313
Prove that $\gcd(f(x),g(x)) = 1$.
55441
Check: For the integers $a,b,c$ show that $\gcd(a,bc)=\gcd(a,\gcd(a,b)\cdot c)$
18772
If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.