Numbers That Won't Split

A baker puts cookies on a tray in neat rows, with the same number in every row. Twelve cookies are easy: 2 rows of 6, 3 rows of 4, or 1 long row of 12. The next day the baker has 13 cookies. 2 rows? One cookie is left over. 3 rows? One left over again. 4, 5, 6 rows? Always something left over. Why does 13 refuse to make a neat rectangle?

Only one rectangle

In the Factors lesson we saw that the factors of a number are the rectangles its dots can make. 12 dots make three rectangles: 1 × 12, 2 × 6 and 3 × 4. So 12 has six factors: 1, 2, 3, 4, 6 and 12.

13 dots make only one rectangle: a single line, 1 × 13. Its only factors are 1 and 13 itself.

Twelve blue dots laid out as three different rectangles: 1 row of 12, 2 rows of 6 and 3 rows of 4. Beside them, thirteen dots: they make only one rectangle, a single line of 13

A number bigger than 1 whose only factors are 1 and itself is called a prime number, or a prime. 13 is a prime. A number bigger than 1 that has more factors, like 12, is called a composite number: it can be built from smaller numbers multiplied together (12 = 3 × 4).

The first primes

Let's check the numbers one by one, looking for a neat rectangle with more than one row:

2: only 1 × 2. Prime.
3: only 1 × 3. Prime.
4: 2 × 2. Composite.
5: prime. 6 = 2 × 3: composite. 7: prime.
8 = 2 × 4, 9 = 3 × 3, 10 = 2 × 5: all composite.
11: prime. 12: composite. 13: prime.

The primes up to 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Notice that 2 is the only even prime. Every other even number is 2 rows of something, so it has 2 as a factor.

The numbers from 1 to 30 in tiles. The primes, 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29, are blue; the composite numbers are grey; 1 is white, on its own

What about 1?

1 dot makes only one rectangle too. But 1 has only one factor, itself, not two. Mathematicians agreed long ago that 1 is not a prime, and not composite either. It sits on its own. This is not just a rule to remember: in the Only One Way to Build It lesson we will see why it matters.

Try it at home

Take 7 buttons. Try to lay them out in 2 rows, 3 rows, and so on. Is 7 prime?
Do the same with 15 buttons.
Is 21 prime? Is 23?

Answers: 1. no rectangle except a line, so 7 is prime; 2. 3 rows of 5, so 15 is composite; 3. 21 = 3 × 7 is composite, 23 is prime.

Words to know

prime number (prime) - a number bigger than 1 whose only factors are 1 and itself, like 2, 3, 5, 7 and 13
composite number - a number bigger than 1 with more than two factors, like 4, 12 and 15

Big idea

A prime number makes only one rectangle of dots, a single line: its only factors are 1 and itself. A composite number can be split into smaller factors. 1 is neither.