A confession and a conjecture $\gcd(a-b,a+b)|2\gcd(a,b)$
38404
Prove $\gcd(a+b, a-b) = 1$ or $2\,$ if $\,\gcd(a,b) = 1$
117098
Properties of the greatest common divisor: $\gcd(a, b) = \gcd(a, b-a)$ and $\gcd(a, b) = \gcd(a, b \text{ mod } a)$
25573
If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.
147527
Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = \gcd(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.
95889
Prove that $\gcd(n! + 1, 3n! + 1) = 1$, $\forall n \in \mathbb{N}$, $n \geq 2$.
69279
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
18714
$\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1$, or $p$
23590
Show that $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n d^{n-1},a-b)$
107095
Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$
20823
If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
51538
Intuition for $\,c\mid a,b \iff c\mid \gcd(a,b)$
101872
Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
19656
Proving that $ \gcd(a,b) = as + bt $, i.e., $ \gcd $ is a linear combination.
2208
$\gcd(a,b,c)=\gcd(\gcd(a,b),c)\,$ [GCD Associative Law]
144571
A conjecture including binomial coefficients
7090
Show that $\gcd(a,bc)=1$ if and only if $\gcd(a,b)=1$ and $\gcd(a,c)=1$