If $\gcd (a,b)=1$ and $a\mid bc$ then $a\mid c$; and if $a\mid c$, $b\mid c$ and $\gcd (a,b)=1$ then $ab\mid c$

Coprime integers: $\gcd (a,b)=1$

$a$ and $b$ are coprime if and only if $ax+by=1$ for some integers $x,y$; and in that case the only common divisors of $a$ and $b$ are $1$ and $-1$

$\gcd$ is symmetric and unchanged by signs: $\gcd (a,b)=\gcd (b,a)=\gcd (\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid })$; moreover $\gcd (a,0)=\mathrm{\mid }a\mathrm{\mid }$, $\gcd (a,1)=1$, $\gcd (a,a)=\mathrm{\mid }a\mathrm{\mid }$, and $\gcd (a,b)\ge 1$ unless $a=b=0$

Common divisor, and the greatest common divisor $\gcd (a,b)$, with the convention $\gcd (0,0):=0$

Divisibility is reflexive and transitive on $\mathbb{Z}$, and is linear: if $d\mid a$ and $d\mid b$ then $d\mid ax+by$ for all integers $x,y$; also $d\mid a$ implies $d\mid ac$, $-d\mid a$ and $d\mid -a$

Divisibility in $\mathbb{Z}$: $d\mid a$ when $a=dq$ for some integer $q$

The integers form a commutative ring

Arithmetic on the integers