So I want to talk about 7, because 7's a bit of a weird number. I mean it's weird because it's hard
to work out when something is divisible by 7 - if you've ever tried to do that you go, oh is that
7 times table? So I wanted to show you one weird trick to work out if something's divisible by
7. You do need to know your 7 times table a little bit - up to like say 100 - but then if you have
a larger number this is a useful trick. So we'll pick a number 434, is it divisible
by 7? I don't know just looking at it, I can't tell, so this is my one weird trick: you split it up into
two parts. I'm going to remove the last digit and we've got the rest of it as 43 and the last digit
as 4. And the trick is this: 5 times the last digit, so it's gonna be 5 times 4, plus the
rest 43, that's the little calculation I'm gonna do. What have I got? I've got 20 plus 43
is equal to 63. If that answer is divisible by 7 your number was divisible by 7. And it is! That is,
what, 9 times 7? So I now know 434 is divisible by 7, that's the little trick. It works the other
way as well, if that had failed the test, if it was not a multiple of 7, then the original number
is not a multiple of 7 either. And you can even do this repeatedly if you've got a really big
number you can actually do this a couple of times. Let's say you got a bigger number, 6468,
Is that divisible by 7? I don't know, I'm going to use my trick. So what I can do is I take
the last digit, 8, gonna multiply that by 5 and I add on the rest 646. What I got here? 40
plus 646, 686. I still don't know if that's divisible by 7 but I can repeat the process, so I'll just go
again. Take the last digit, so 6 times 5, and add the rest, the 68. well if I got? 30 plus 68, it's
98. Do you know what I could stop there, I know 98 is divisible by 7, but just for fun let's take it
one extra step further, maybe you don't know 98 is divisible by 7. Last digit is 8, times 5 plus
the rest which would just be a 9 and that would be 40 plus 9 is 49. Yeah I'm confident that's
divisible by 7, it's 7 squared; and therefore my original number, which was the big number the 6468,
it's a multiple of 7. - (Brady: What do you do if you do the process to 49?) - Oh okay okay let's find
out, I don't know, I haven't done it! Let's find out. 49 uh okay last digit is a 9 times 5
plus the rest of it which is just the 4; and now what I'm gonna get here? 9 times
5 is 45 - tell me if I'm wrong - I forgot that, oh plus 4, oh it's 49.
Oh dear, we've got ourselves into a loop. These numbers can get larger because I'm adding
with this test, I'm adding a number on like so they kind of get big. There is an alternative version
we could do which might be more useful because the numbers get smaller instead. I do split it up again,
there's the 43 and the 4. I take the rest as I was calling it, the 43, I subtract
two times the last digit this time so this time it would be 43 minus 8 which would be 35.
Is that divisible by 7? Yes it is, therefore my number was. So that's a subtracting version of it
which might be more useful in some situations, maybe less useful in other situations. Do you
want to hear why it's true? You might want to try and work out why it's true actually, um but I am
going to show you so if you want to work it out make sure you do that, I'm going to show you how
it works. Why is that true? So if I write a number I could write a number like this: as 10x plus y,
y is the last digit and this 10x is the rest - or the x is what I was calling the rest. That's
just a different way of writing down a number. So I want to know if this is divisible by 7. I hope
you would agree that if if this is divisible by 7, 5 times the number is divisible by 7 as well. I
think that's true isn't it? Yeah. So let's multiply this by 5. So I'm going to multiply this by 5,
so what am I going to get? 50x plus 5y. So I might want to know, is that divisible by 7? I'm going to
split this up in a different way, I'm going to have a little bracket here and I'm going to have x
plus 5y in a bracket and what's left over? I guess it's all the x's I haven't used, there's 49x left over.
So I want to know is this divisible by 7? 49 x is, definitely, I know that's definitely divisible by
7. So really the only thing that I need to ask is is that bracket is divisible by 7? But that
bracket is my test. This is my 5 times the last digit plus the rest. So if that bracket passes the
test, if it's divisible by 7, the original number was divisible by 7. And if it's not
then the original number wasn't either. - (Is it ever important to know if a number is divisible by 7?)
I- I'm sure I can think of situations where you would need to find if something is divisible by 7.
I mean it's more useful to know when things are difficult by 4 or 2 or
6 is kind of useful; and 7 is less useful unless you go into like a restaurant and you've got
7 people and you want to split the bill, you know, that kind of thing, so I can imagine. - (Rugby maybe?)
(Converted tries in rugby? Or or a touchdown with a field goal- with a conversion? A)
(touchdown with a conversion - 7 points? Yeah?) - Look at my face - look how blank
my face is now, I have no idea what you're talking about. But maths is good isn't it?
Now that we've seen a test for the divisibility of 7, maybe it's nice to do a recap of some of the
tests for smaller numbers. And they might be useful, keep them in your pocket and then you'll know them.
Right how to tell if your number is divisible: 7 we know, we've now got a test for this, and it was the
5 times last digit plus rest. Awesome, tick. 1, well everything's divisible by 1, okay
I'll just write that in then, everything. Let's do the easy ones next; divisible by 2, even
right? Even numbers. But another way of saying even is they end with a 0, a 2 a 4 or an 8,
and then you know it's- - (Or a 6) - And 5 is an easy one as well isn't it? Ends with a
0 or 5? Yep. And 10's an easy one, ends with a 0, cool. Those are the easy ones. The
next easiest test I think it's divisible by 9, I've actually done a video about divisible by 9.
A trick that can check your arithmetic but I think this is something that everyone should know... - Before
we called it casting out 9s or the rule of 9; and the test for this is something called
the digit sum. We add up all the digits and if the digit sum is divisible by 9, tick, the original
number was. Let's just do an example. I'll steal the example I did before, right, 6468.
So the digit sum would be adding up the digits 6 plus 4 plus 6 plus
8. What have I got there? Uh 10, 16, 24. Which I guess means it's not divisible by 9, that failed
the divisibility test, but it does pass the test for 3. The test for 3 is the digit sum
again, this time if the digit sum is divisible by 3 the original number was divisible by
3. 11? Shall we have a look at the test for 11? That's the alternating sum. It's the same
idea really, you alternate the plus and minuses; let's do that example to show you what I mean.
So I had this number again, the the 6468, start with the 6,
you subtract the 4, you add the 6, you subtract an 8. So you keep plus and minusing the digits
like that. If that's 0 or a multiple of 11 it passes the test. What if I got here?
It's a 2, an 8 minus an 8 is a 0. That's okay, so if it's a 0 it passes the test,
that means it's a multiple of 11. My number was a multiple of 11, all right. So what have we got left?
The next easiest one is 6 - how do we know if it's divisible by 6? Well all you do is
you apply two tests. Does it pass divisibility by 2 and the visibility by 3? If it does
then it's divisible by 6. So you have to pass two tests: divisibility by 2 and 3. What about
my example I've been using, the 6468? Yeah it passed the 3 test and it passes the even test
so it's divisible by 6. You do the 12 the same way, so it just needs to pass divisibility by
3 and 4. Oh I haven't done 4 so I should mention it; 4, what is the divisibility by 4
test? You look at the last two digits. Let's use- let's use my example here. I just care about the
68. I really don't care how big this number is either, it's just it's just the 68 I care about,
and then is that divisible by 4? You might think, but is that hard? Working out if that's divisible by
4? I like to do um can I divide by 2 twice? Is it doubly even? So is it- can I divide by 2
and then is that even? I can divide by 2 again. So if I did that 68, is it doubly even? It'd be 34,
and then yeah that's even, which would be a 17 if I halved it again. That passes. And for 8:
this is the- apart from 7 I'd say 8 is that maybe the harder one.
It's the last three digits; don't care how big the number is but it's the last three digits.
And I'll do my little example: here's my 6468, I only care about the end of it, the 468 part, there
it is - is that divisible by 8? That's again kind of hard, is it triply even then? So is it
even? Can I divide by 2? Can I divide by 2? Can I divide by 2? So a little harder, but I think you
could do that in your head. Let's try it: 468, 234 for halving it, and then can I halve it again? 117,
so I've halved it once, I've halved it twice, I think I'm gonna get stuck. So it's not divisible by
8 because it wasn't triply even. - This video was supported by Jane Street; a research driven trading
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Divisibility - I thought it would just be nice to do a recap of um divisibly-
say that again! I thought it'd just be nice to do a recap of disaviliby - oh!
(This is turning into good bonus material!) - I've discovered a word I can't say