Lowest common multiple (LCM)

It is dark, and you are standing on a headland. Two lighthouses out at sea. The near one flashes every eighteen seconds. The far one keeps its own rhythm, every twenty four. Right now, by chance, they flash at the same instant. Then they drift apart. One gets ahead, the other lags, and the pattern looks like it will never line up again. But it will. Sooner or later both timers land on the same second. When? That is a real question, and it is the same question when two buses leave the stop together, when two sets of traffic lights change together, when two planets come back into line. Anything running on its own repeating cycle. The number you are looking for has a name. It is the lowest common multiple.

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

And to find one, you need almost nothing new. The same two numbers from the video on highest common factor, ninety and one hundred and twenty six. The same two factor trees. Just one thing changes, right at the end.

Before any method though, take this with you. A common factor divides into your numbers, so it can never be bigger than the smaller one. A common multiple is what they divide into, so it can never be smaller than the bigger one.

On a number line, they live in different places. For ninety and one hundred and twenty six, the highest common factor sits at ninety or below. The lowest common multiple sits at one hundred and twenty six or above. They can never be the same answer, and that check stays in the corner for the rest of this video.

Where these came from, quickly. Factor trees, from the video on prime factorisation. Ninety: nine and ten, down to three, three, two and five. One hundred and twenty six: two and sixty three, then nine and seven, then three and three.

So ninety is two, times three squared, times five. One hundred and twenty six is two, times three squared, times seven. Two identical starting lines. Everything now depends on what you take from them.

For the highest common factor, you take only the overlap, the two and the two threes that both numbers own, and multiply just those. Eighteen. Smaller than both, exactly as promised.

For the lowest common multiple, you do the opposite. Take every prime that appears in either number, at the highest power it reaches anywhere. Nothing is left outside.

Why the highest power? Because your answer must be a multiple of both numbers, so it has to contain each one completely. Whichever number demands more copies of a prime, the answer carries that many.

So a two, a three squared, a five and a seven. Two times nine is eighteen, times five is ninety, times seven is six hundred and thirty.

Six hundred and thirty. Bigger than one hundred and twenty six, as it had to be. Now hear the two rules together. Highest common factor: shared only, lowest power. Lowest common multiple: everything, highest power.

One piece of exam craft before we go on. This is what students actually wrote on a Foundation paper. The examiner's report says: twelve and twenty four were common wrong answers. Not silly answers. Those students found factors, checked them, took the biggest one. Real work, real numbers. But they were highest common factors, and the question had asked for the lowest common multiple. Examiners see the same swap again and again. Which is exactly what the size check is for. Both of those answers were smaller than the numbers they came from, and a lowest common multiple never is. One glance at the corner of the page catches it.

So, quick check. Twenty four and thirty six. Without calculating anything, what do you already know about their highest common factor, and their lowest common multiple? Have a think. I'll give you five seconds.

The highest common factor must be twenty four or less. The lowest common multiple must be thirty six or more. You knew both of those before doing a single calculation.

Now, back to the headland. One lighthouse every eighteen seconds, the other every twenty four, both flashing right now. You have everything you need for this.

Notice that nothing in it says lowest common multiple. But things happening together again always mean it. You want the first moment that both eighteen and twenty four divide into.

You can simply list. Eighteen, thirty six, fifty four, seventy two. Twenty four, forty eight, seventy two. Seventy two seconds is the first moment they share.

Primes get there too, and they are safer when the numbers are large. Eighteen is two times three squared. Twenty four is two cubed times three. The powers differ here, so take the highest of each: two cubed, and three squared. Eight nines are seventy two.

And here is the slip. Take the lower power of each instead, and you get two times three, which is six. Six is smaller than both numbers. That is the highest common factor wearing the wrong label, and the size check catches it instantly.

Your turn. Find the lowest common multiple of six and eight. Remember what your answer must be at least as big as. Off you go. Take your time with it. I'll wait, and then we will work it through together.

Six is two times three. Eight is two cubed. Highest power of each: two cubed, times three. Eight threes are twenty four. Bigger than eight, so it passes the check.

So, let's pull it together. Every prime that appears, at its highest power, multiplied. Or list multiples until one is shared. Then check the size. And when you meet adding fractions, this is the number you will be hunting for as the common denominator.

That was video three of five. Next in the chapter: systematic listing strategies.

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