Foundations — Factors and multiples — all factors of a number, factor pairs

This page assumes nothing. We build every symbol the parent note uses, one brick at a time, in the exact order they depend on each other.

1. The dot picture — what "a number" even means here

Before any symbol, start with something you can hold: a pile of identical objects. We will draw them as dots.

The number $5$ just means "this many dots":

2. The multiplication sign $\times$ — repeated grouping

Look at $3\times 4$: three rows, four dots in each row.

WHAT we just did: turned a symbol into a rectangle. WHY: the whole topic of factors is secretly the topic of "which rectangles can I build from $n$ dots?" The multiplication sign is the door into that idea. WHAT IT LOOKS LIKE: a filled grid — count the dots and you get $12$.

3. The division sign $÷$ and "remainder" — sharing out

Sometimes it works cleanly. Sometimes dots are left over — those leftovers are the remainder.

WHY the topic needs remainder: the entire definition of a factor hangs on this one word. $a$ is a factor of $n$ only when the remainder is $0$ — when the rectangle comes out perfectly full, no lonely dots.

4. Letters that stand for numbers — $a$, $b$, $k$, $n$

The parent note writes $n=a\times k$. Those letters scare people. They shouldn't.

Letters let us talk about every number at once instead of redrawing the argument for $12$, then $24$, then $36$. That is why the parent note derives its rule with $a$ and $b$ rather than a specific number.

5. Factor and multiple — the two names for one rectangle

Now every ingredient is on the table, so the parent's core definitions are free.

6. Ordered pair notation $(a,b)$ — a factor pair

WHY the topic needs it: every rectangle you can build hands you two factors at once (both sides). Listing factor pairs is the fast way to list every factor without missing one.

7. Square root $\sqrt{n}$ and the floor $\lfloor \cdot \rfloor$ — the stopping line

This is the most feared symbol on the parent page, so we build it slowly.

WHY THIS tool and not another? We need a fair "middle" of every rectangle. In any rectangle with sides $a$ and $b$ where $a\le b$, the short side $a$ can be at most the side of the square — because if $a$ grew past $\sqrt{n}$, so would $b$, and the two together would overflow $n$. The square root is exactly the fence between "short side" and "long side". That is why it, and nothing else, is the stopping line.

WHAT IT LOOKS LIKE (above): two rectangles for $n=12$. The short sides $1,2,3$ all sit at or left of the dashed square-root line $\sqrt{12}\approx 3.46$; their long partners $12,6,4$ sit to the right. No pair has both sides past the line — that is the whole reason we stop scanning at $\lfloor \sqrt{n}\rfloor$.

8. The inequality signs $\le$, $<$, $>$ — comparing sizes

The parent uses these when it argues "$a\times b>n$".

WHY the topic needs them: the whole proof that "one side is $\le \sqrt{n}$" is an argument about which is bigger. Without $\le$ and $>$ you cannot even state the reason the $\sqrt{n}$ rule works.

9. The "if and only if" arrow $⟺$

Prerequisite map

Each foundation above feeds directly into the parent topic and its neighbours: Prime numbers and composite numbers, Prime factorisation, HCF and LCM, Divisibility rules, Perfect squares, and Fractions in lowest terms.

Equipment checklist

Hide the right side and test yourself — if you can answer all, you're ready for the parent note.