When Do They Meet Again?

At a bus station in Nairobi, two buses leave together at 8:00. The blue bus leaves every 4 minutes, and the green bus every 6 minutes. When do they next leave at the same time?

Two lists of jumps

The blue bus leaves at 8:00, then 4 minutes later, then 4 more... Its times after 8:00 are the multiples of 4, as in the Multiples lesson:

blue bus: 4, 8, 12, 16, 20, 24 minutes after 8:00, ...
green bus: 6, 12, 18, 24 minutes after 8:00, ...

The numbers in both lists are 12, 24, and so on. They are the common multiples of 4 and 6. The first one, the smallest, is 12. So the buses next leave together at 8:12, and again at 8:24.

A number line from 0 to 24. Blue jumps of 4 go above it and green jumps of 6 go below it. They land on the same number at 12 and at 24

The least common multiple

The smallest number that is a multiple of two numbers is called their least common multiple. Some books call it the lowest common multiple: it is the same thing. The least common multiple of 4 and 6 is 12.

It answers "when do they meet again?" and also "how many do I need so it comes out even?" Sausages come in packs of 6, and bread rolls in packs of 8. You want one sausage in each roll, with none of either left over. How few can you buy?

packs of sausages give 6, 12, 18, 24, 30, ...
packs of rolls give 8, 16, 24, 32, ...

The least common multiple of 6 and 8 is 24: buy 4 packs of sausages (4 × 6 = 24) and 3 packs of rolls (3 × 8 = 24).

Four packs of 6 sausages next to three packs of 8 bread rolls: 24 sausages and 24 rolls, one sausage for each roll

A quicker way to find it

You don't have to write two long lists. Take the multiples of the bigger number, one by one, and stop at the first one the smaller number divides exactly:

for 6 and 8: 8? 6 doesn't divide it. 16? No. 24? Yes: 24 ÷ 6 = 4. So the answer is 24.

Multiplying the two numbers always gives a common multiple: 6 × 8 = 48. But it is not always the least one. It is the least only when the two numbers have no common factor except 1, like 3 and 5: their least common multiple is 3 × 5 = 15. When they share a factor, as 6 and 8 share 2, they meet sooner.

Try it at home

Clap every 2 counts and stamp every 3 counts while you count to 12. On which counts do you clap and stamp together?
What is the least common multiple of 3 and 4? Of 4 and 10?
One light flashes every 5 seconds and another every 2. If they flash together now, when do they next flash together?

Answers: 1. 6 and 12. 2. 12; 20. 3. After 10 seconds.

Words to know

common multiple - a number that is a multiple of two numbers at once, like 24 for 4 and 6
least common multiple (or lowest common multiple) - the smallest common multiple of two numbers

Big idea

Things that repeat every 4 and every 6 meet again at the common multiples of 4 and 6. The first meeting is the least common multiple, 12. It is the product of the two numbers only when they share no factor but 1.