Prime factorisation, and why there is only one of them per number

0:00A teacher asks the class a question. Are fifty-six and sixty-three co-prime? The first student splits them up. Fifty-six is fourteen times four. Sixty-three is twenty-one times three. Then she looks at her four numbers. Fourteen, four, twenty-one, three. Nothing repeated. So she says yes, they're co-prime. Every bit of that arithmetic is correct. Fourteen times four really is fifty-six. But the answer is wrong. Can you see where it went wrong, before I show you?

0:38A second student splits the same two numbers differently. Fifty-six is seven times eight. Sixty-three is nine times seven. And there it is. Seven, in both of them. Seven divides fifty-six, and seven divides sixty-three. They share a factor, so they are not co-prime. So what went wrong the first time? Nothing, in the arithmetic. The mistake was thinking that one factor pair shows you all the factors. It doesn't. Fourteen times four tells you two things divide fifty-six. It says nothing about the other four factors.

1:18The seven was hiding inside the fourteen the whole time. So the obvious fix is to try harder. Find more factor pairs. Eighty is forty times two. And twenty times four. And ten times eight. And sixteen times five. Four different splittings. And the textbook puts a row of question marks after them, which is exactly right. Because you still can't be sure. How do you know you're finished? Maybe there's a fifth splitting you haven't found, hiding a factor you haven't seen.

1:55More factorisations doesn't mean more certainty. It just means more guessing. We need something that finishes. Something that tells you when to stop. Here it is. Don't split once. Keep splitting. Fifty-six is four times fourteen. Now break the pieces. Fourteen is two times seven. So fifty-six is four times two times seven. Four is two times two. So fifty-six is two times two times two times seven. And now look at what we're holding. Two, two, two and seven.

2:33Can we break a two? No. Can we break a seven? No. They're prime. So we stop. That's the whole method, and there's your hidden seven, out in the open. The stopping rule is the good part, and it needs no judgement at all. You stop when everything you're holding is prime. And that isn't a choice. A prime has exactly two factors, one and itself. So the only way to split a prime is one times itself, and that gets you nowhere.

3:06A prime has nothing left to give. The splitting stops because it can't continue. That's why this method always finishes, and finding factor pairs never does. Now, two words that sound the same and mean different things. Worth thirty seconds. The prime factorisation of fifty-six is two times two times two times seven. That's the whole expression. The factorisation. But the prime factors of fifty-six are just two and seven. Only two of them.

3:42The two appears three times in the factorisation, but two is still one prime factor. So fifty-six has four numbers written down, and two prime factors. The factorisation is the recipe. The prime factors are the ingredients you need. Two edge cases, and the book settles both on the spot. First. What's the prime factorisation of a prime, like seven? It's just seven. One prime, all by itself. That counts.

4:15Second, and this one matters more. What about one? One has no prime factorisation at all. Not one. Not nothing. It simply doesn't have one. And you can see why we need that rule. If one were allowed in, you could write fifty-six as two times two times two times seven times one. And then times one again. And again, forever. No factorisation would ever be finished. So one is kept out, and the method terminates.

4:48Now the beautiful part. Does the route you take change the answer? Let's take thirty-six, and walk into it by four different doors. Door one. Thirty-six is two times eighteen. Eighteen is two times nine. Nine is three times three. Two, two, three, three. Door two. Thirty-six is three times twelve. Twelve is three times four. Four is two times two. Three, three, two, two. The same four numbers. Door three. Four times nine. Four is two times two, nine is three times three.

5:27Two, two, three, three. Again. Door four. Six times six. Each six is two times three. Two, three, two, three. And that's the same collection one more time. Four completely different routes. Two twos and two threes, every single time. That's not a coincidence, and it's the reason this whole idea is worth anything. Now, the four doors came out in different orders. Does order matter? It doesn't, and here's a way to see it that has nothing to do with primes.

6:04Here's a block of cubes. Three wide, five deep, two tall. Count it in layers going down. Two layers, each three by five, so two times fifteen. Thirty. Now count it in slabs from the front. Five slabs, each three by two, so five times six. Thirty. Now from the side. Three slices, each five by two, so three times ten. Thirty. Same block. Same cubes. We just counted them in a different direction.

6:37Multiplication doesn't care what order you go in, because you're describing the same pile either way. The textbook names this properly, calls it commutativity and associativity, and saves it for a later class. Since the order carries no information, we may as well agree on one. The usual habit is smallest first. Two hundred and twenty-five is three times three times five times five. Thirty is two times three times five.

7:11And that's a convention, not a rule. If you write three times two times five, you are not wrong. You've just written it in an unusual order. The mathematics is identical. So far we've been breaking downward. You can also build upward, and it's often faster. Take seventy-two. Split it once, into twelve times six. Now, you may already know these. Twelve is two times two times three. Six is two times three.

7:44So put them together. Seventy-two is two times two times three, times two times three. Tidy that into increasing order. Two times two times two times three times three. And here's the check that makes it trustworthy. Count the twos. Twelve brought two of them, six brought one. Three twos in the answer. It agrees. Count the threes. Twelve brought one, six brought one. Two threes in the answer. It agrees.

8:16Nothing appeared from nowhere, and nothing went missing. And now the payoff. You can factorise things without multiplying them out. What's the prime factorisation of fifty-six times twenty-five? Don't multiply. Fifty-six is two times two times two times seven. Twenty-five is five times five. So the answer is just both lists, poured together. Two, two, two, five, five, seven. We never worked out that fifty-six times twenty-five is one thousand four hundred. We didn't need to.

8:55Try it yourself on one thousand times eighty-one. And here's what all of this was for. Go back to the very first question. Fifty-six is two, two, two, seven. Sixty-three is three, three, seven. Both lists contain a seven. Not co-prime. Certain, this time, not guessed. The prime factorisation is a fingerprint. Every number has exactly one, and no two different numbers share theirs. That's a big claim, and here's my question for you. Why should it be true?

9:30Why can't some number have two completely different prime factorisations? Tell me what you think in the comments.