FALSE: For all integers $a$ and $b$, $\gcd (a,b)\cdot \mathrm{lcm}(a,b)=ab$

Every common multiple of $a$ and $b$ is a multiple of $\mathrm{lcm}(a,b)$, and $\gcd (a,b)\cdot \mathrm{lcm}(a,b)=\mathrm{\mid }ab\mathrm{\mid }$

Common multiple, and the least common multiple $\mathrm{lcm}(a,b)$, taken to be $0$ when $a=0$ or $b=0$

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

Every common divisor of $a$ and $b$ divides $\gcd (a,b)$; consequently $d=\gcd (a,b)$ exactly when $d\ge 0$, $d\mid a$, $d\mid b$, and every common divisor of $a$ and $b$ divides $d$ — a characterisation that holds at $(a,b)=(0,0)$ as well

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 absolute value $\mathrm{\mid }a\mathrm{\mid }$ of an integer

Absolute value in $\mathbb{Z}$: $\mathrm{\mid }a\mathrm{\mid }\ge 0$; $\mathrm{\mid }a\mathrm{\mid }=0$ exactly when $a=0$; $\mathrm{\mid }-a\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }$; $\mathrm{\mid }ab\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }$; $-\mathrm{\mid }a\mathrm{\mid }\le a\le \mathrm{\mid }a\mathrm{\mid }$; and $\mathrm{\mid }a\mathrm{\mid }\le c$ exactly when $-c\le a\le c$

The integers have no zero divisors; multiplicative cancellation

The integers form a commutative ring

The integers form a totally ordered ring

Arithmetic on the integers

Order on the integers

The naturals embed in the integers

The integers as equivalence classes of pairs of naturals