What HCF and LCM do for consecutive, even, and co-prime numbers

0:00Take six and eighteen, and find their highest common factor. It is six. Which is odd, because six was one of the numbers we started with. The answer did not come out somewhere in between. It came out as one of the pair. So find more pairs that do that. Two and four. Two and six. Three and twelve. Five and twenty. Now the real question. What do all of those have in common?

0:29In every one, the smaller number divides the larger exactly. And it is exactly those. Whenever the smaller divides the larger the answer is the smaller one, and whenever it does not, it is not. But we have checked four pairs, and there are infinitely many. So here is the move that turns four into all of them. Call the smaller number n. Then the larger one has to be a multiple of n.

1:00Two n, or five n, or a hundred n. Take n and five n. What is their highest common factor? It is n. And you can see why without trying a single example. n divides n, and n divides five n, so n is a common factor. And nothing bigger than n divides n. That one letter did what a hundred examples could not. It covered every case at once.

1:29Now two numbers standing next to each other. Eight and nine share nothing. Forty-four and forty-five share nothing. You might think that big enough neighbours would eventually share something. Four thousand nine hundred and ninety-eight, and four thousand nine hundred and ninety-nine. Nothing. And here is the reason, which is better than any number of examples. Suppose some number divides both of them. Then it divides the gap between them too.

2:02The gap between neighbours is one. And nothing bigger than one divides one. So there is nothing to find, at any size. Which also means their lowest common multiple is always the two multiplied together. Now even numbers, where a wrong turn is waiting. Two consecutive even numbers. Six and eight give two. Fourteen and sixteen give two. It is always two. Two consecutive odd numbers. Nine and eleven give one. It is always one.

2:38So what about two even numbers, any two? Two and four give two. Four and six give two. Six and eight give two. Three for three, so it must be two. Eight and twelve give four. Twelve and eighteen give six. Twenty and thirty give ten. Not a rare accident either: more than a third of even pairs break it. The honest statement is that two even numbers have an even highest common factor. Exactly two is a claim about consecutive evens, and that is a stronger thing to say.

3:16Some pairs share nothing at all. Those have a name: co-prime. It does not mean either of them is prime. Eight and nine are co-prime, and neither is prime. Eight is three twos, nine is two threes. They have no prime in common, so the only thing dividing both is one. And something falls out of that immediately. If nothing is shared, then nothing gets counted twice, so their lowest common multiple is the whole product.

3:48Eight and nine give seventy-two, which is eight times nine. In fact the highest common factor times the lowest common multiple is the product, for every pair there is. If the first is one, the second has to be everything. Now ask the opening question again, but of the lowest common multiple. When does it land on one of the two numbers? Three and twenty-four. The answer is twenty-four, which is one of the pair.

4:21And it is the same condition as before: the smaller has to divide the larger. Which means when that happens, both answers land on the pair at once. The highest common factor is the smaller number, and the lowest common multiple is the larger. Back to n and five n. The highest common factor is n, and the lowest common multiple is five n. One pair, both extremes, and neither of them anywhere new.

4:54Four more, and each has a reason rather than a rule to remember. Two numbers next to each other. Nothing shared, so the lowest common multiple is the whole product. Two co-prime numbers. Nothing shared, so the product again. Same reason, not a second fact. Two consecutive even numbers. Six and eight. They share a two and only a two, so one of those twos gets counted twice in the product. The lowest common multiple is half of it: twenty-four, not forty-eight.

5:31And two multiples of three. Six and nine give eighteen. Six and twelve give twelve. Eighteen is three sixes and twelve is three fours, so no fixed answer is hiding here. All you can honestly say is that the result is a multiple of three. An experiment. Take a pair, double both numbers, and watch the highest common factor. Two hundred and seventy is two, three, three, three, five. Fifty is two, five, five.

6:05They share a two and a five, so the answer is ten. Now double both. Five hundred and forty, and a hundred. Five hundred and forty is two, two, three, three, three, five. A hundred is two, two, five, five. Each picked up exactly one more two. So the shared part picked up exactly one more two. Ten becomes twenty. The highest common factor doubled, and it had no choice.

6:34And it is not special to doubling. Multiply both by anything you like and the highest common factor is multiplied by the same thing. Which brings us to what this whole idea is really about. Two numbers, written as products so you can see inside them. Fourteen times six, and fourteen times nine. Both are multiples of fourteen, so fourteen is a common factor. Is it the highest one? Break them into primes and look. Fourteen times six is two, two, three, seven. Fourteen times nine is two, three, three, seven.

7:16There is the two and there is the seven, and together they make the fourteen we already knew about. But look again. There is a three in both rows as well. So the highest common factor is two, three and seven — forty-two, not fourteen. And that extra three came from the six and the nine. So when is a shared multiplier the whole answer? Eighteen times ten, and eighteen times fifteen. The leftovers are ten and fifteen, and they share a five.

7:51So the answer is not eighteen. It is eighteen fives: ninety. Ten times thirty-eight, and ten times twenty-one. The leftovers are thirty-eight and twenty-one, and those share nothing. So this time the answer really is ten. Five times thirteen against five times twenty. Thirteen and twenty share nothing, so the answer is five. Twelve times sixteen against twelve times twenty. Sixteen and twenty share a four, so the answer is twelve fours: forty-eight.

8:27There is the condition. The shared multiplier is the whole answer exactly when what is left over shares nothing. Look back at all of that and it is one sentence. Multiply both numbers by something, and the highest common factor is multiplied by that same something. Doubling was that sentence with the something set to two. n and five n was that sentence with n out front, and one and five left over, which share nothing.

9:01And the trap was the same sentence with fourteen out front, and six and nine left over — which do share something. There is no list of special cases here. There is one line: the smaller-count rule with the shared part pulled out in front. Every pattern here is that line read in a different direction. Which is the whole point of finding the reason instead of collecting more examples.

9:30One last thing, about patterns and facts. Two and four give two. Four and six give two. Six and eight give two. Three examples, all agreeing, and the claim is still false. Eight and twelve give four. Agreeing examples can never finish the job. You can only ever check finitely many. One disagreeing example finishes it instantly, the other way. So here is one to try. Find two numbers with highest common factor one and lowest common multiple sixty-six.

10:06Sixty-six is two, three and eleven. Nothing is shared, so between them the pair uses each of those primes exactly once. Split three primes into two piles and there are four ways: one and sixty-six, two and thirty-three, three and twenty-two, six and eleven. And you can say there are no others — because you did not find them, you built them.