Math Core

Some numbers can be arranged into lots of different rectangles. Others can only make one long, skinny row. That difference sorts every whole number bigger than 1 into two groups: prime and composite.

One rectangle or many?

Try to arrange 7 tiles into a rectangle with more than one row. You can't. The only way is a single row:

🟩🟩🟩🟩🟩🟩🟩

That is because the only factor pair of 7 is $1 \times 7$ .

Now try 6 tiles. You can make one row of 6, or 2 rows of 3:

🟧🟧🟧

6 has the factor pairs $1 \times 6$ and $2 \times 3$ . It has more than two factors.

Definition

Prime and composite

A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7.

A composite number has more than two factors. Examples: 4, 6, 8, 9.

What about 1?

The number 1 has only one factor: 1. That is not exactly two, and it is not more than two. So 1 is neither prime nor composite. It is in a group all by itself.

The primes up to 50

Here are all the prime numbers from 1 to 50:

2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19,\ 23,\ 29,\ 31,\ 37,\ 41,\ 43,\ 47

The prime numbers up to 30 are marked with dots.

Notice that 2 is the only even prime. Every other even number has 2 as a factor, as well as 1 and itself, so it has at least three factors.

How to decide

To tell whether a number is prime or composite, look for one factor other than 1 and the number itself.

Found one? The number is composite.
Can't find any? The number is prime.

Tip

Quick checks for a factor you can spot fast:

Ends in 0, 2, 4, 6 or 8 → 2 is a factor.
Ends in 0 or 5 → 5 is a factor.
Digits add up to a multiple of 3 → 3 is a factor. (For 51: $5 + 1 = 6$ , so 3 is a factor.)

Worked example: Prime or composite: 15

Is 15 prime or composite?

15 ends in 5, so 5 is a factor: $3 \times 5 = 15$ . It has factors 1, 3, 5, 15. That is more than two, so 15 is composite.

Worked example: Prime or composite: 23

Is 23 prime or composite?

23 is odd, so 2 is not a factor.
$2 + 3 = 5$ , not a multiple of 3, so 3 is not a factor.
4 can't be a factor, since 23 is odd.
23 doesn't end in 0 or 5, so 5 is not a factor.

You can stop here: $5 \times 5 = 25$ is already bigger than 23, so any bigger factor would need a partner smaller than 5, and you tested all of those. So 23 has only 1 and 23 as factors. It is prime.

Worked example: A tricky one: 51

Is 51 prime or composite?

It looks prime, but $5 + 1 = 6$ , which is a multiple of 3. Divide: $51 \div 3 = 17$ . So $3 \times 17 = 51$ . 51 is composite.

Common mistake

Odd does not mean prime. 9, 15, 21, 25, 27 and 49 are all odd, but they are composite ( $9 = 3 \times 3$ , $49 = 7 \times 7$ ). Always look for a factor before you decide.

Practice

Practice 1

Which number is prime?

Practice 2

Is 21 prime or composite?

Practice 3

Which statement about the number 1 is true?

Practice 4

List the two prime numbers between 30 and 40.

Separate answers with commas, e.g. 2, -5

Practice 5

57 is composite. Find its smallest factor other than 1.

Type a number.

Practice 6

Which list contains only composite numbers?

Practice 7

How many prime numbers are there between 40 and 50?

Type a number.

Practice 8

Mia is thinking of a prime number between 50 and 60. Its digits add up to 8. What is her number?

Type a number.