Factors: Making Rectangles

A teacher in Hanoi has 12 chairs for a reading circle. She wants them in equal rows, with no chair left over and no gap. How many ways can she set them out?

Every rectangle of 12

Try each row length and see whether the chairs fill the rows exactly:

1 row of 12, or 12 rows of 1;
2 rows of 6, or 6 rows of 2;
3 rows of 4, or 4 rows of 3.

Rows of 5 don't work: two rows of 5 use 10 chairs and 2 are left over. Rows of 7, 8, 9, 10 or 11 don't work either.

The three rectangles 12 chairs can make: 1 row of 12, 2 rows of 6, and 3 rows of 4. Under each one is its multiplication: 1 × 12, 2 × 6 and 3 × 4

The numbers that make a rectangle of 12 are called the factors of 12:

1, 2, 3, 4, 6 and 12.

A factor of 12 is a number you can multiply by another number to get 12. Each rectangle gives two factors at once: 3 rows of 4 shows that 3 and 4 are factors. So factors come in factor pairs: 1 and 12, 2 and 6, 3 and 4.

Factors and multiples are two sides of one fact. 3 × 4 = 12 says that 3 is a factor of 12, and also that 12 is a multiple of 3 (it is where the jumps of 3 land after four jumps).

Finding all the factors

To find every factor of 36 without missing one, go up from 1 and write each pair:

1 × 36
2 × 18
3 × 12
4 × 9
5 doesn't work (36 ÷ 5 leaves 1 over)
6 × 6

Now stop. The next pair would be 9 × 4, and that is 4 × 9 again, turned round. The pairs meet in the middle, at 6 × 6. So the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

The factors of 36 written in a row, from 1 to 36, with arcs joining each factor pair: 1 with 36, 2 with 18, 3 with 12, 4 with 9, and 6 joined to itself in the middle

Many factors, or few

Every number has at least two factors: 1 and the number itself (one long row). Some have many more: 12 has six factors, 36 has nine. But try 7 chairs: rows of 2, 3, 4, 5 or 6 all leave chairs over. Seven chairs only make one long line. Numbers like that are special, and a whole module of this course is about them.

Try it at home

Take 18 buttons and find every rectangle you can make. Write the factors of 18.
Find the factors of 20 by going up from 1. Where do the pairs meet?

Answers: 1. 1 × 18, 2 × 9, 3 × 6, so the factors are 1, 2, 3, 6, 9, 18. 2. 1 × 20, 2 × 10, 4 × 5 (3 doesn't work); after 4 × 5 comes 5 × 4, so the factors are 1, 2, 4, 5, 10, 20.

Words to know

factor - a number that multiplies with another number to make a given number: 3 is a factor of 12, because 3 × 4 = 12
factor pair - two factors that multiply to make the number, like 3 and 4 for 12

Big idea

The factors of a number are the row lengths that make a rectangle of it. They come in pairs, and the pairs meet in the middle, so you can find them all by going up from 1 until the pairs turn round.