Factors and Multiples
Sharing 12 cookies fairly is easy — into groups of
2, 3, 4 or
6 — but 12 will never split into
5 equal piles. The numbers that do divide something evenly,
and the numbers its times table lands on, are exactly what we are about to meet.
Two words come up again and again once you start
multiplying:
factors and multiples. They are really two views of
the same idea — one looks inside a number, the other looks onward from it.
A factor of a number divides it exactly — it goes in a whole number
of times with nothing left over. Think of factors as the numbers that fit
into a number perfectly. The factors of 12 are:
1,\ 2,\ 3,\ 4,\ 6,\ 12
Check one: does 3 divide 12 exactly?
Yes — 12 \div 3 = 4, no remainder, so 3 is
a factor. Does 5? No — 12 \div 5 = 2
remainder 2, so 5 is not a factor
of 12.
A multiple of a number is what you get by multiplying it by
1, 2, 3, \dots — it is simply that number's times table.
Multiples come out of a number. The multiples of 3 are:
3,\ 6,\ 9,\ 12,\ 15,\ 18,\ \dots
So 12 appears in both lists: it is a multiple of
3, and 3 is a factor of
12. That is not a coincidence — it is the whole secret, and we
come back to it at the end.
Find factors by making rectangles
Here is the picture worth keeping. If you can lay out a number of dots in a
perfect rectangle — equal rows, no gaps, none left over — then the number of
rows and the number in each row are both factors of that number. Such a tidy
grid is called an array.
Take 12 cookies. You could arrange them as
3 rows of 4, or
2 rows of 6, or one long row of
12. Every rectangle that works names a pair of factors — and a
number with lots of rectangles has lots of factors.
That is 12 cookies set out as 3 rows of
4. The tray is full and square — nothing hangs off the edge — so
3 \times 4 = 12, and both 3 and
4 are factors of 12. Slide the same
cookies into 2 rows of 6 and you have
found another factor pair without baking a single extra cookie.
Factors come in pairs
Every rectangle has two side lengths, so factors always arrive two at a time.
That is why the easiest way to list all the factors of a number is to hunt for its
factor pairs — each pair multiplies to give the number:
12 = 1 \times 12 = 2 \times 6 = 3 \times 4
Read the pairs off and you have collected every factor:
1, 2, 3, 4, 6, 12. Notice you can stop searching once the two sides
meet in the middle — past 3 \times 4 the pairs just repeat the other
way round (4 \times 3, 6 \times 2, …).
Press play, then step through. Each step lays out one rectangle for the chosen number, and the
two side lengths are a factor pair — because that side length
divides the
number exactly. Press Refresh for a brand-new number to break apart.
Factors stop; multiples go on forever
A big difference is worth saying out loud. A number has only a handful of factors
— you can write them all down and the list ends — because nothing bigger than the number
can divide into it. But a number's multiples never stop: keep adding the number
and the times table marches on without end.
- Factors of 12: 1, 2, 3, 4, 6, 12
— exactly six of them, and that is the lot.
- Multiples of 12: 12, 24, 36, 48, 60, \dots
— and the dots really do go on for ever.
A handy memory hook: factors are never bigger than the number (the biggest factor
of 12 is 12 itself), while
multiples are never smaller (the smallest multiple of
12 is 12).
Worked examples
1. List the factors of 18. Hunt for factor pairs:
18 = 1 \times 18 = 2 \times 9 = 3 \times 6
After 3 \times 6 the sides would cross over, so we stop. The factors are
1, 2, 3, 6, 9, 18.
2. Is 7 a factor of 20?
Try it: 20 \div 7 = 2 remainder 6. There is a
leftover, so no — 7 is not a factor of
20. (You could not make a perfect rectangle of
20 cookies with rows of 7.)
3. Write the first five multiples of 5. Count up the
five times table:
5,\ 10,\ 15,\ 20,\ 25
Every one ends in 0 or 5 — a neat
fingerprint of the multiples of 5.
The two classic mix-ups:
- Don't swap them round. Factors go into a number and are
never bigger than it; multiples come out of a number and are
never smaller. "Factor of 12" must be one of
1,2,3,4,6,12; "multiple of 12" must be
12,24,36,\dots
- 1 and the number itself are always factors. Every number divides by
1 and by itself, so those two are always on the factor
list — don't forget them when you count.
Try to set out 7 cookies in equal rows of more than one and someone
always ends up holding a spare. The only rectangles are 1 row of
7 (or 7 rows of 1)
— so the only factors of 7 are 1
and 7. A number whose only factors are 1
and itself is called a prime number.
Two sides of the same coin
Factors and multiples are tied together by one clean rule. Saying
3 is a factor of 12 is
exactly the same statement as saying 12 is a multiple of
3:
b \text{ is a factor of } a \iff a \text{ is a multiple of } b
Both just mean a = b \times k for some whole number
k. So you never really learn two things here — you learn one
relationship and name it from whichever end you are standing at.
Khan Academy walks through finding factors and multiples here: