Divisibility: Definitions and Basic Properties

Try this

1. Drag $d$ slowly from $1$ to $12$. Which values of $d$ make the remainder $0$? Compare them with the row of “Divisors of $24$” below.
2. Fix $d=6$ and increase $n$ one step at a time. How often does “rows all full” show up?
3. Set $n$ to $13$, then to $1$. How many values of $d$ still give a rectangle?
4. With $n=24$, $d=4$ gives $6$ rows and $d=6$ gives $4$ rows: divisors always come in pairs. Set $n$ to $36$. Which $d$ is paired with itself?

Making a rectangle is exactly what the following definition says.

Definition Let $a$ and $b$ be integers with $a\ne 0$. If there is an integer $q$ such that $b=aq$, we say that $a$ divides $b$ and write $a\mid b$. Then $a$ is a divisor of $b$, and $b$ is a multiple of $a$. If no rectangle is possible, we write $a\nmid b$.

Even when no rectangle is possible, things don't get messy: you can always fill $q$ full rows, with $r$ dots left over, where $0\le r<a$. This is division with remainder, $b=aq+r$. Divisibility is just the case where the remainder is $r=0$.

Why

Why is the sum of two multiples of $d$ still a multiple of $d$? Stack the two piles of dots and look.

Two more properties that the “rows of dots” picture makes clear:

• Transitivity: if $a\mid b$ and $b\mid c$, then $a\mid c$. (Rows of $a$ fill $b$ exactly, and $c$ is made of several copies of $b$.)
• Size: if $a\mid b$ and $b\ne 0$, then $|a|\le |b|$. So a nonzero integer has only finitely many divisors. This is the lifeline for solving Diophantine equations later on.

Remove one condition

Practice