🌲 HCF and LCM: Which One When?

Hello! There are two kinds of problem. In one, we look for the biggest tile. In the other, we wait for two buses to meet again. Both are solved with factors and multiples. But which one, and when?

A board is 24 centimetres by 36 centimetres. We will cover it with equal square tiles, with no cutting and no gaps. If we try tiles 8 centimetres wide, a strip is left over. Tiles 12 centimetres wide fit exactly. The tile length must divide both 24 and 36.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. The ones that are in both lists are the common factors. The biggest of them is 12. It is called the HCF: the highest common factor.

With big numbers, writing lists takes a long time. Instead we use a prime factor tree. 24 equals 2 times 2 times 2 times 3. 36 equals 2 times 2 times 3 times 3. The leaves that both trees share are 2, 2 and 3. We multiply them: the HCF equals 12.

Now the second problem. Two buses leave the bus stop together. Bus A comes every 8 minutes, and bus B comes every 12 minutes. A comes at minutes 8, 16 and 24. B comes at minutes 12 and 24. They are together again at minute 24. This is called the LCM: the lowest common multiple.

We can find the LCM with the trees too. 8 equals 2 times 2 times 2. 12 equals 2 times 2 times 3. The two 2s they share, we take once. Then we add what is left over: one more 2 from 8, and a 3 from 12. 2 times 2 times 2 times 3 equals 24.

Some numbers have no common prime factor at all. 8 equals 2 times 2 times 2, and 9 equals 3 times 3. There is no common factor, so the HCF is 1. Numbers like these are called coprime. The numbers themselves do not have to be prime! In this case the LCM is the two numbers multiplied: 72.

In a test, how do you know which one to use? Words like sharing out, dividing, cutting and biggest square say HCF. Words like together again, first time and how many minutes later say LCM. The HCF is never bigger than the smaller number. The LCM is never smaller than the bigger number.

Here is a little way to check. For two numbers, the HCF times the LCM equals the two numbers multiplied together. 24 times 36 equals 864. 12 times 72 is 864 too. You can check your answers like this.

Summary: the HCF is the product of the common prime factors, and it appears in sharing and cutting problems. The LCM is the product of all the factors, with the shared ones taken once, and it appears in together-again problems. If the HCF is 1, the numbers are coprime. Now try it in Power Telescope!