$1+ab$ is a unit if and only if $1+ba$ is a unit.
83255
Finding inverse in non-commutative ring
5223
$\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$
7090
Show that $\gcd(a,bc)=1$ if and only if $\gcd(a,b)=1$ and $\gcd(a,c)=1$
76680
Prove that if $\gcd(a,c)=1$ and $\gcd(b,c)=1$ then $\gcd(ab,c)=1$
140252
Matrices - Conditions for $AB+BA=0$
12954
Prove if $ac>bc$ and $c>0$, then $a>b$
20878
If $(a,b)=1$ then prove $(a+b, ab)=1$.
1991
If $\gcd(a,b)=1$ and $a$ and $b$ divide $c$, then so does $ab$
53113
$I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible
76049
Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$
1448
$x^a - 1$ divides $x^b - 1$ if and only if $a$ divides $b$
91506
If $\gcd(a,b) = 1$ then $\gcd(a,a + b) = 1$
56399
Proof for $A \cup B = B$ if and only if $A \subset B$
57125
Prove the distributive law $a(b+c)=ab+ac$ for real numbers?
70307
$(A\cap B)\cup C = A \cap (B\cup C)$ if and only if $C \subset A$
96027
Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite.
123555
Proving that if $ab=e$ then $ba=e$