Multiples: Counting in Jumps

A market stall in Lima sells eggs only in boxes of 6. You can buy one box, or two, or three - but you can't open a box. Can you buy exactly 20 eggs?

Counting in jumps

Count the eggs box by box:

1 box: 6 eggs;
2 boxes: 12 eggs;
3 boxes: 18 eggs;
4 boxes: 24 eggs.

The number of eggs goes up by 6 each time: 6, 12, 18, 24, ... So 20 is never reached. After 18 the next number is 24, and 20 is jumped over.

The numbers we land on are called the multiples of 6. A multiple of 6 is 6 times some number: 6 × 1 = 6, 6 × 2 = 12, 6 × 3 = 18, and so on.

You can see the same idea on a number line. A frog starts at 0 and jumps 3 each time. It lands on 3, 6, 9, 12, 15, ... These are the multiples of 3. Every number it jumps over, like 7 or 11, is not a multiple of 3.

A frog jumping along a number line from 0, 3 steps at a time. It lands on 3, 6, 9, 12, 15 and 18, which are circled. The numbers in between are jumped over

Two more things about multiples:

They never end. The frog can always make one more jump. There is no biggest multiple of 3.
0 is a multiple of every number, because 6 × 0 = 0: zero boxes, zero eggs. The frog starts there before its first jump.

Is it a multiple?

To check whether 20 is a multiple of 6, you don't have to count all the way up. Divide: 20 ÷ 6 is 3 remainder 2 (remainders were in Numbers Level 1). There is something left over, so 20 is not a multiple of 6. But 24 ÷ 6 = 4 with nothing left over, so 24 is a multiple of 6.

Patterns in the multiples

Write the numbers from 1 to 50 in rows of ten, and colour the multiples. The multiples of 5 make two straight columns: every one ends in 5 or 0. The multiples of 9 make a slanting line, because each one is 10 more and 1 less than the one before.

Two grids of the numbers 1 to 50 in rows of ten. In the first, the multiples of 5 are coloured and make two straight columns. In the second, the multiples of 9 are coloured and make a slanting line

Patterns like these are a first hint that multiples follow rules. In the next module we turn them into quick tests.

Try it at home

Write the first ten multiples of 4. Which of them are also multiples of 8?
Buttons come in packs of 5. Can you buy exactly 35? Exactly 42?

Answers: 1. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40; every second one (8, 16, 24, 32, 40) is a multiple of 8. 2. 35 yes (7 packs); 42 no, it is between 40 and 45.

Words to know

multiple - what you get by multiplying a number by 0, 1, 2, 3, and so on: the multiples of 6 are 0, 6, 12, 18, 24, ...

Big idea

The multiples of a number are where you land counting in jumps of that number. They never end, 0 is always one of them, and a number is a multiple when dividing it leaves nothing over.