Divisibility
Divisibility is the property of one integer being divided by another with no remainder; divisibility rules provide shortcuts for checking this.
Formula
a \text{ is divisible by } b \text{ if } a = b \times k \text{ for some integer } k
Definition
A number is divisible by another if dividing them leaves no remainder; formally, $a$ is divisible by $b$ (written $b \mid a$) if there exists an integer $k$ such that $a = bk$. Divisibility rules give quick tricks to check this without dividing: by $2$ (even last digit), by $3$ (digit sum divisible by $3$), by $4$ (last two digits divisible by $4$), by $5$ (ends in $0$ or $5$), by $9$ (digit sum divisible by $9$), and by $11$ (alternating digit sum divisible by $11$), rules that derive from the fact that $10 = 9+1$ (so $10^k \equiv 1 \pmod 9$) and $10 = 11-1$ (so $10^k \equiv (-1)^k \pmod{11}$). In $\mathbb{Z}$, divisibility defines a partial order, $b \mid a$ iff $b\mathbb{Z}$ contains $a\mathbb{Z}$ as ideals, making $\mathbb{Z}^+$ a distributive lattice under GCD (meet) and LCM (join); in any integral domain divisibility is a preorder, refining to a lattice in a UFD.
Example
A number is divisible by $2$ if it ends in $0$, $2$, $4$, $6$, or $8$, so $348$ is divisible by $2$; a number is divisible by $5$ if it ends in $0$ or $5$. Is $2{,}346$ divisible by $6$? It must be divisible by both $2$ and $3$: the last digit $6$ confirms divisibility by $2$, and the digit sum $2+3+4+6=15$ confirms divisibility by $3$, so yes. For primes without a simple rule, like $7$, repeatedly apply $2a_0 - b$ (where $b$ is the remaining digits and $a_0$ the last digit) until small: for $343$, $34 - 2(3) = 28$, which is divisible by $7$, so $343 = 7^3$ is too.
Key Insight
Divisibility rules save time: you can tell $312$ is divisible by $3$ just by adding $3+1+2=6$, and since $6$ is divisible by $3$, so is $312$. The divisibility relation in $\mathbb{Z}$ is the starting point for the theory of ideals; Dedekind's insight was that in rings of algebraic integers, unique prime ideal factorization restores the divisibility structure lost by the failure of unique element factorization.