Sharing and meeting: meet the HCF and LCM

Here are two puzzles that look completely different.

Puzzle 1. Meera's grandmother wants to tile a courtyard that is 240 cm long and 180 cm wide with square tiles, all the same size, with no cutting and no gaps. She wants the tiles to be as big as possible, so there are fewer joins to clean. How big should each tile be?

Puzzle 2. At Diwali, two strings of lights are switched on at the same moment. One string flashes every 4 seconds, the other every 6 seconds. After how many seconds do they flash together again?

The first puzzle is about cutting something into equal pieces. The second is about two repeating things meeting again. By the end of this layer you will be able to solve both in your head, and you will know the two special numbers hiding inside them: the HCF and the LCM.

Take 12 marbles. Can you put them into equal rows with none left over?

• 1 row of 12 ✓
• 2 rows of 6 ✓
• 3 rows of 4 ✓
• 4 rows of 3 ✓
• 6 rows of 2 ✓
• 12 rows of 1 ✓
• 5 rows? 12 ÷ 5 = 2 remainder 2. ✗ Two marbles are left over.

The numbers that worked, 1, 2, 3, 4, 6, 12, are the factors of 12. A factor of a number divides it exactly, with no remainder.

Notice they came in pairs: 1 and 12, 2 and 6, 3 and 4. Each pair multiplies to 12. That pairing trick is the fastest way to list factors without missing any.

Now take two numbers, 12 and 18, and write their factors side by side.

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18

Some numbers appear in both lists: 1, 2, 3, 6. These are the common factors of 12 and 18. "Common" here means shared, just as a common room is a room everyone shares.

The highest (biggest) of the common factors is 6. It is called the Highest Common Factor, or HCF, of 12 and 18.

We write this as HCF(12, 18) = 6.

The HCF appears whenever you want to split things into equal groups or equal pieces, as big as possible, with nothing left over.

The key words are largest, greatest, maximum, longest, as many as possible (identical groups), and a feeling of cutting, dividing, sharing or packing. The answer is always smaller than or equal to the numbers you started with, because you are breaking them down.

Now for the other idea. Count in 4s: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, … These are the multiples of 4. They are just the 4 times table, and they go on for ever.

A multiple of a number is what you get when you multiply it by 1, 2, 3, 4, and so on. So 20 is a multiple of 4 (because 4 × 5 = 20), but 22 is not.

Factors and multiples are two sides of one fact. 4 × 5 = 20 tells you that 4 and 5 are factors of 20, and that 20 is a multiple of 4 and of 5.

Write the multiples of 4 and of 6 side by side:

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …

Some numbers are on both trails: 12, 24, 36, … These are the common multiples of 4 and 6. Unlike common factors, they never run out: 12, 24, 36, 48, 60 and so on.

The lowest (smallest) common multiple is 12. It is called the Lowest Common Multiple, or LCM. We write LCM(4, 6) = 12.

Look at the list of common multiples again: 12, 24, 36, 48… They are all multiples of 12. Once you know the LCM, every other common multiple is just a multiple of it.

The LCM appears whenever two or more repeating things start together and you want to know when they will next line up: lights flashing, bells ringing, buses leaving, runners lapping, or just the smallest number that several numbers all divide.

The key words are together again, at the same time, next, first time, smallest or least (number that…), and a feeling of repeating, cycles and meeting. The answer is always bigger than or equal to the numbers you started with, because you are waiting for both cycles to finish.

Put the two answers for 12 and 18 next to each other:

• HCF(12, 18) = 6. It is at most the smaller number, 12.
• LCM(12, 18) = 36. It is at least the bigger number, 18.

This gives you a quick sense check. If you are asked for an HCF and your answer is bigger than one of the numbers, something went wrong. If you are asked for an LCM and your answer is smaller than one of the numbers, something went wrong.

The HCF divides each number. Each number divides the LCM. So the HCF always sits at the bottom and the LCM at the top, with the numbers themselves in between.

Most word problems about HCF and LCM are easy once you decide which one you need. Ask yourself one question:

Am I breaking things into equal parts, or waiting for repeating things to line up?

If you are breaking, sharing, cutting, packing or arranging into the biggest equal groups, you want the HCF. The answer will be smaller than the numbers.

If you are waiting for things to happen together, or looking for the smallest number that several numbers all divide, you want the LCM. The answer will be bigger than the numbers.

The best way to get comfortable is to meet lots of problems. For each one below, first decide: sharing (HCF) or meeting (LCM)? Then find the answer by listing, and finally do the sense check: an HCF answer is small, an LCM answer is big.