$360=2^3\cdot 3^2\cdot 5$ and $84=2^2\cdot 3\cdot 7$, with $\gcd (360,84)=12$ and $\mathrm{lcm}(360,84)=2520$ read off the exponents

For positive integers $a$ and $b$ and every prime $p$: $v_p(\gcd (a,b))=\min \{v_p(a),v_p(b)\}$ and $v_p(\mathrm{lcm}(a,b))=\max \{v_p(a),v_p(b)\}$; so the exponent-wise greatest common divisor is the $\gcd$ of the divisibility page and not a second notion

For $n\ge 1$ and any injective list $p:r\to \mathbb{Z}$ of primes containing every prime divisor of $n$, one has $n={\prod }_{i<r}{p}_i^{v_{p_i}(n)}$; the exponents are determined by $n$, and $v_q(n)=0$ for every prime $q$ outside the list

$v_p(ab)=v_p(a)+v_p(b)$ for nonzero integers $a,b$, and $v_p(a+b)\ge \min \{v_p(a),v_p(b)\}$ whenever $a$, $b$ and $a+b$ are all nonzero

For a prime $p$ and a nonzero integer $a$: $p^{v_p(a)}\mid a$ and $p^{v_p(a)+1}\nmid a$; $p^k\mid a$ holds exactly for $k\le v_p(a)$; $v_p(a)\ge 1$ exactly when $p\mid a$; $v_p(1)=v_p(-1)=0$; and $v_p(p)=1$

The $p$-adic valuation $v_p(a)$ of a nonzero integer: the greatest $k\in \mathbb{N}$ with $p^k\mid a$

The fundamental theorem of arithmetic: every integer $n\ge 1$ is a product of primes, and the factorisation is unique up to order — if ${\prod }_{i<r}{p}_i={\prod }_{j<s}{q}_j$ with every $p_i$ and $q_j$ prime, then $r=s$ and $q_i=p_{\pi (i)}$ for some $\pi \in \mathrm{Sym}(r)$

Every integer $n\ge 1$ is a finite product of primes: there are $r\in \mathbb{N}$ and a list $p:r\to \mathbb{Z}$ of primes with $n={\prod }_{i<r}{p}_i$, the case $n=1$ being the empty product

Prime and composite integers: $p$ is prime when $p>1$ and its only positive divisors are $1$ and $p$

The Euclidean algorithm: for $a\in \mathbb{Z}$ and $b>0$ the remainder descent from $(a,b)$ terminates, and the last nonzero remainder is $\gcd (a,b)$

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

Division with remainder in $\mathbb{Z}$: for $a\in \mathbb{Z}$ and $b>0$ there are unique $q,r\in \mathbb{Z}$ with $a=qb+r$ and $0\le r<b$

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

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 $d\mid -a$

If $d\mid a$ and $a\mathrm{\ne }0$ then $d\mathrm{\ne }0$ and $\mathrm{\mid }d\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }$; hence the set of divisors of a nonzero integer is bounded above by $\mathrm{\mid }a\mathrm{\mid }$

Euclid's lemma: if $p$ is prime and $p\mid ab$ then $p\mid a$ or $p\mid b$

Powers $g^n$: natural exponents in a monoid and integer exponents in a group, with $g^0=e$

Exponent laws in a group: $g^{m+n}=g^m{g}^n$ and $(g^m{)}^n=g^{mn}$ for all $m,n\in \mathbb{Z}$, and $(gh{)}^n=g^n{h}^n$ when $g$ and $h$ commute

$(\mathbb{Z},\cdot ,1)$ is a commutative monoid whose group of units is $\{1,-1\}$; equivalently $u\mid 1$ holds exactly for $u=1$ and $u=-1$

Semigroup and monoid

The product $g_0{g}_1\cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n=0$) equal to the identity

The integers form a commutative ring

The integers form a totally ordered ring

Arithmetic on the integers

Order on the integers

The integers as equivalence classes of pairs of naturals

The naturals embed in the integers