Divisibility of Integers

I always get this confused, but $5\mid 10$ and $10\nmid 5$. 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 $m\mid n$ (”$m$ 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 $m\nmid n$.

Transitivity of Divisibility (TD)

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

Divisibility of Integer Combinations (DIC)

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

Other

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