Given that 𝑎 and 𝑏 are positive integers with $𝑎\mid𝑏$, how would one prove that $\gcd(𝑎, 𝑏) = 𝑎$?
31465
Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
25573
If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.
78678
If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$
69279
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
19656
Proving that $ \gcd(a,b) = as + bt $, i.e., $ \gcd $ is a linear combination.
36
Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$
33996
Proving that $\gcd(2^m - 1, 2^n - 1) = 2^{\gcd(m,n )} - 1$
74631
For natural numbers $a$ and $b$, show that $a \Bbb Z + b \Bbb Z = \gcd(a, b)\Bbb Z $
21788
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
77278
If $\,\gcd(a,b) = {\rm lcm}(a,b)\,$ then prove $a=b$
72747
Proving gcd($a,b$)lcm($a,b$) = $|ab|$
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
7151
Let $a,m,n \in \mathbf{N}$. Show that if $\gcd(m,n)=1$, then $\gcd(a,mn)=\gcd(a,m)\cdot\gcd(a,n)$.
34330
How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
55673
Alternate proof that $\exists\ x \in \Bbb{Z}$ such that $ \gcd (a+bx,c) = 1$?
25907
Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
136064
How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?