The four divisibility facts you can prove, and how to use them

0:00Add two even numbers together. The answer is even. Everybody knows that one. So here is a harder question. Is the answer a multiple of four? 12 and 16 make 28. Four goes into 28 exactly. Yes. 12 and 6 make 18. Four does not go into 18. No. So sometimes, which is the most annoying answer in mathematics. It means there is a rule and you have not found it yet.

0:33Out of 1681 pairs of even numbers, 841 add up to a multiple of four. A shade over half. Something is deciding which ones. Let us go and find it. Take an even number and lay it out in rows of four. 12 goes in exactly. Three complete rows, and nothing left over. 18 does not. Four complete rows, and two dots sitting on their own at the end. And two is the only thing that can ever be left over. An even number cannot leave one, and it cannot leave three.

1:14So there are exactly two kinds of even number, and only two. The multiples of four, and the multiples of four with two extra. Write them as four p, and four p plus two. Every even number there has ever been is one or the other. There is no third kind to worry about. Three cases, then. The first is the easy one. Take two numbers of the first kind. 12 and 16.

1:45Slide the second block underneath the first one. Every row was complete before, and every row is still complete now. Three rows and four rows make seven rows. 28. In writing, four p plus four q is four times p plus q. 16 and 28 behave the same way. Four rows and seven rows make eleven. 44. The picture and the algebra are not two different arguments. They are one argument, told twice.

2:20Second case, and this is the one that surprises people. Take two numbers of the second kind. 6 and 10. Neither one of them is a multiple of four. Each block has complete rows, and then two dots left over on their own. Slide them together. And now look at what happens at the bottom. The two short rows meet, and two dots plus two dots is four dots. That is a complete row.

2:49So the answer has four complete rows. 16. The two leftovers did not spoil it, they finished each other off. That extra row is the plus one in four p plus q plus one. 2 and 6 make 8. 22 and 6 make 28. Every single time. Third case. One of each kind. 12 and 6. The first block is exact. The second has two dots spare. Slide them together, and those two dots are still spare. There is nothing for them to pair up with.

3:26Four complete rows, and two over. 18. And it fails in exactly the same way every time. Always two over. Never one, never three, never anything else. Of the 840 mixed pairs, not a single one lands on a multiple of four. So there is the rule we were looking for. Same kind, yes. Different kinds, no. Stop for a moment, because something happened there that is easy to walk straight past.

3:59We did not check every pair of even numbers. We checked three shapes. And the blocks were never given a size. p and q stood for any number of rows at all. So those three pictures cover every pair of even numbers there has ever been, and every pair there ever will be. Which is worth saying plainly. The drawing is not an illustration of the proof. The drawing is the proof.

4:26The algebra sitting next to it is saying the same thing in a different alphabet. And that one move, rearranging complete rows, has four separate facts inside it. Let us go and collect them. First fact. If a number divides two others, then it divides their sum. Eight divides 16, and eight divides 56. Two complete rows of eight, and seven complete rows of eight. Slide them together and you have nine complete rows. 72.

5:01Run the same argument backwards. Take the smaller block away instead of adding it, and the rows that are left are still complete. So it divides the difference too. 80 and 120 do the same thing. Ten rows and fifteen rows. 200. Across 3971 combinations of row length and row count, it never once fails, and nothing in the argument ever asked what the numbers were. But be very careful which way round you read that.

5:39Eight divides 72. Does that mean eight divides the two numbers you added together to get 72? 48 and 24. Eight divides both of those. It worked. 50 and 22. Eight divides neither of those. And they still add to 72. There are 71 ways to split 72 into two parts. Only 8 of them have both parts divisible by eight. The fact travels from the parts up to the total. It does not travel back down again.

6:15And on a total eight does not divide, like 74, it comes apart completely. 18 of the splits have one part divisible by eight, and not one single split has both. Second fact. Every multiple of a multiple is still a multiple. Seven divides 14. Now take 14 and repeat it five times over. Each copy is two complete rows of seven, so five copies is ten complete rows. 70. And ten rows of seven is exactly what 70 is.

6:53Repeating a block cannot possibly spoil a row, so this fact comes for free. Now the trap, and it is the mirror image of the last one. Divisible by seven does not mean divisible by fourteen, or twenty-one, or twenty-eight. 42 is divisible by seven. It is divisible by fourteen. It is not divisible by twenty-eight. Of the first twelve multiples of seven, only four of them go into 42.

7:27Third fact. A divisor drags every one of its own factors along behind it. Twelve divides 60. And twelve is two sixes, and twelve is also three fours. So cut every row of twelve in half, and you have rows of six. Cut each row into three, and you have rows of four. A factor of the row length cuts every row cleanly, so the whole block stays covered. Twelve has six factors. One, two, three, four, six and twelve. All six of them divide every multiple of twelve.

8:05Drawn as rings, every multiple of 32 sits inside the multiples of eight, and those sit inside the multiples of four. But it never travels outward. Of the 124 multiples of eight below a thousand, only 31 are multiples of 32. Fourth fact, and this one is a different animal. Ask for two row shapes at the same time. A number divisible by nine and by four. The smallest is 36, and every such number turns out to be a multiple of 36.

8:45Nine times four is also 36, so it is very tempting to say: just multiply the two divisors together. So try that. Divisible by six, and divisible by four. Six times four is 24. But look at 12. It is divisible by six. It is divisible by four. And it is not divisible by 24. Of the 33 multiples of twelve below 400, only 16 are multiples of 24. The product rule gets it wrong nearly half the time.

9:23And the reason is sitting in plain sight. Six and four both contain a two, and multiplying them counts that same two twice over. So the answer is the smallest number that both of them divide, and not their product. For six and four, that is 12. The product is only right when the two divisors share nothing at all. Of 361 pairs, 216 share nothing, and those are exactly the 216 where multiplying works.

10:00Nine and four share nothing, which is why 36 came out right both ways. That was luck, and not a rule. Here are the four, then. Every multiple of a multiple. Every factor of a divisor. The sum and the difference of two multiples. And two divisors at once, forcing the smallest number they both divide. The first three are complete rows being slid about, and you can prove all three with a drawing. The fourth one is not, and it needs an argument of its own.

10:35Which is the whole lesson. Knowing why a rule is true is the only thing that tells you how far it reaches.