Highest common factor (HCF)

A rectangle of wall. Ninety centimetres one way, one hundred and twenty six the other. You are covering it in square tiles. No cutting, no gaps. Six centimetre tiles fit both sides exactly. But that is three hundred and fifteen tiles. You want the biggest tile that still fits. Ten centimetres? Ninety splits into nine of them. One hundred and twenty six does not split at all. You would be cutting, and guessing could take all day. There is a biggest tile that works, and a way to find it with no guessing. It is called the highest common factor.

This is video two of five in our chapter on factors, multiples and counting.

First, what the words mean. A factor divides into a number exactly. A common factor divides into both of your numbers. The highest common factor is the biggest one that does.

One warning before any working starts. This needs factors, not multiples. If you catch yourself writing out a times table, you have built the wrong list. Examiners report it every year. Their words: it is clear many candidates cannot distinguish between HCF and LCM, with many writing out multiples. That other list gives the lowest common multiple, the next video, not this one.

Quick one, so you can check yourself. Which of these is a common factor of twelve and eighteen? Four, six, or twenty four?

Have a look. I'll give you a few seconds.

The answer is six. Six divides into twelve, and into eighteen. Four goes into twelve but not eighteen. And twenty four is bigger than both, so it divides into neither.

That last one is worth keeping. A factor is never bigger than the number it came from. So your highest common factor can never be bigger than your smaller number. A free check, every time.

Back to the wall. Find the highest common factor of ninety and one hundred and twenty six, showing enough working to justify it.

That part is factor tree work, covered in the video on prime factorisation. Ninety is two, times three squared, times five. One hundred and twenty six is two, times three squared, times seven.

Give each number its own circle. Ninety's primes on the left. One hundred and twenty six's on the right.

Anything both numbers own moves into the overlap. Both own a two. Both own two threes. The five belongs to ninety alone, the seven to one hundred and twenty six. Those stay outside.

Now the step that costs people marks. Do it deliberately. Middle only, then multiply. Take what is in the overlap, and nothing from outside.

Here is why that gives you the biggest one. Any number that divides into both can only be built from primes that both numbers own. Those primes are exactly what is in the middle. Use every one, and nothing larger exists.

So, two, times three, times three. Two nines are eighteen. The highest common factor of ninety and one hundred and twenty six is eighteen.

Size check: eighteen is smaller than ninety, so it passes. And your wall takes eighteen centimetre tiles, five across and seven down. Look where that five and seven came from. The primes you left outside.

Now the working, because eighteen on its own can score nothing, even when eighteen is right. The examiner's report is explicit. In order to score both marks there had to be a clear demonstration of why this figure was identified as the HCF, either by multiplying common prime factors or having a complete list of six factors for each number. So write the multiplication down. Two times three times three equals eighteen. That line is the demonstration. Or give a complete factor list, all of it, not the first few. Calculators too. A different report: some students used their calculators to find the answer of thirty six directly with no working. A factorise button is not creditable. The written method is what is marked.

One more, and nothing tells you what to do. A caterer has forty eight mini quiches and eighty four sausage rolls. Identical platters, none left over, as many platters as possible.

Nothing in that says highest common factor. So what is it actually asking you to find?

Your turn to spot it. I'll wait.

The number of platters must divide into forty eight exactly, and into eighty four exactly, and be as big as possible. That is the highest common factor in disguise.

The other route is a full list. In pairs, so you miss none. Forty eight: one and forty eight, two and twenty four, three and sixteen, four and twelve, six and eight. Eighty four the same way, six pairs, down to seven and twelve. Both lists complete, nothing skipped.

Same move as before. What sits in both lists? One, two, three, four, six and twelve. The highest of those is twelve.

Twelve platters. Forty eight quiches between twelve is four each. Eighty four sausage rolls is seven each. Four quiches and seven sausage rolls on every platter.

Your turn, properly. Find the highest common factor of thirty six and sixty. Circle only the primes that appear in both, multiply just those, and check against thirty six. Off you go. Have a go at all three steps. I'll wait, and then we will work it through together.

Thirty six is two squared times three squared. Sixty is two squared times three times five. Both own two twos, so two twos go into the middle. The threes are different here. Thirty six has two, sixty has only one. You can only share what both of them have, so one three goes in. Two, two, three. Twelve. Smaller than thirty six, so it passes.

So, let's pull it together. Factors, never multiples. Build the trees or the lists in full. Then extract deliberately: middle only, then multiply. Write that multiplication down, because the working is the thing being marked. Then check your answer is smaller than your smaller number.

That was video two of five. Next in the chapter: lowest common multiple.

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