# Divisibility of Integers

I always get this confused, but $5∣10$ and $10∤5$. 5 divides 10, 10 doesn’t divide 5.

- I guess now intuitively, it makes sense. 5 divides 10 because you can split 10 into 2 5s

Divisiblity Definition

We say that $m∣n$ (”$m$ divides $n$”) if there exists an integer $k$ so that $km=n$

If $m∣n$, then we say that

- $m$ is a divisor or a factor of n
- $n$ is a multiple of / divisble by $m$
If $m$ does not divide $n$, then we write $m∤n$.

### Transitivity of Divisibility (TD)

For all integers $a$, $b$ and $c$, if $a∣b$ and $b∣c$, then $a∣c$.

### Divisibility of Integer Combinations (DIC)

For all integers $a$, $b$ and $c$, if $a∣b$ and $a∣c$, then for all integers $x$ and $y$, $a∣(bx+cy)$.

### Other

For all integers $a$, $b$ and $c$, if $a∣b$ or $a∣c$, then $a∣bc$.