Hi everyone. So today I've got a question from the 2006 Junior competition and 0%
of students got this one right. So it goes like this: it's quite surprising
that 1 times 2 times 3 times 4 times 5 times 6 equals 8 times 9 times 10. That's
quite an interesting coincidence. Now, what's even more interesting is that we
can do the same thing with 8 times 9 times 10 times 11 times 12 times 13
times 14; that can also be written in a similar way to this, as a product of
consecutive whole numbers. We just got to work out what those numbers are
that go here. So pause the video here and see if you can figure it out. Now before
I go through how I solved that question, first I want to show you the way that I
think about numbers and split them apart and recombine those parts in new ways.
And this is useful in a lot of different situations. For example, if I were to
multiply 15 by 18 in my head, I wouldn't use the traditional procedure to do that.
What I would do is imagine the 18 splitting apart into 2 times 9 and then
I would combine that 2 with the 15 to make 30, and then I've got 30 times 9 and
that's much easier to work with because I know that 9 times 3 is 27 so 9 times
30 is going to be 270. So the reason that that works is that we can do
multiplications in any order you like. So I can think of that as the 18 is 2
times 9, but I don't have to do it in that order; I don't have to do the 2
times 9 first and then multiply by 15. What I can do is do the 15 times 2
first, and then multiply that by 9. So that makes 30, and then you've got 30
times 9. So I want to just give you this visual representation of how I think
about these kinds of things, the way that you can split numbers up
and then recombine those factors in new ways. Another situation where I would use
this style of thinking is when I'm trying to work out the lowest common
multiple of two numbers. So for example if I was working out 3 over 14 plus 5
over 24 and I want to make a common denominator so I can add these fractions.
What I would do is I know that the 14 can be split up into two by seven like
that and then the 24 can also be split up into 2 by 12, so then what I would
do is I would imagine if I multiply this by 7 and this one by 12, you're
actually going to have the same thing in both denominators, because you've got 2
by 7 by 12 there and 2 by 12 by 7 there and they're the same thing just in
different orders. So that gives it a common denominator. Now if I've
multiplied the bottom by 12 here I gotta multiply the top by 12 as well and same
thing over here, we've multiplied by 7, so gotta multiply the top by 7 as well. So
that gives us 36 plus 35 over... and then here we've got like 24 by 7 which is 168,
so that gives us 71 over 168. So I know that these are the same thing because
what you can do is imagine you've got 2 by 12 by 7; you can express that as 24
times 7 which was what I had here or you can think of it as 12 times 14 which is
what I had there. Another situation where I would use this trick is if I'm trying
to factorise a quadratic like this. So don't worry if you haven't studied this
before, but the idea is just that we have to work out two numbers that multiply
together to make 144 and add together to make 25. So the way that I would think
automatically of splitting up 144 is that it's 12 times 12, so let's try that.
But 12 and 12 doesn't work because those add up to 24 and we want them to add up
to 25, so what I might do though is just sort of think of splitting up these
numbers and recombining them. So I would think of 12 as 3 times 4 and then maybe
we could try combining the 3 with the 12 to make 36, but 36 plus 4, that doesn't
work either because that makes 40. So what I might do is split up the 36 into
4 times 9 and then I might try combining that 4 with that one to make 16 and now
that does work: 9 plus 16 makes 25. So that's how I actually do these. I've seen
most of my students sort of try, you know, listing stuff out, like okay they'll do...
if you halve that you get 72, and then they'll try... okay it must be 3
times something, and then like they try to divide that by 3 and it's just... it
takes a bit longer because you have to do all these divisions whereas
recombining the factors is actually a lot easier. Okay now back to the original
question. So the point that I'm trying to make is that we don't actually have to
work out what 8 times 9 times 10 times and so on is in order to be able to
factorise it in a new way. What we can do is just take these numbers and pull them
apart and recombine the factors. So for example I could turn that 12 into 3
times 4 and then I might combine that 4 with that 11
and make 44, so 44 is a factor of this big huge number. And let's just...
let's just play around with these and sort of pull things apart and see what
happens. A lot of these numbers can be pulled into smaller ones so like 9 is 3
times 3, 8 is 2 times 4, and then you can also pull apart the 4 into 2 times 2.
This is actually the same sort of idea as you might have done with factor trees,
where you just split the numbers into smaller numbers. So we can pull apart a
lot of these except for... the 13 and the 11 are both primes, meaning that they
can't be pulled apart into smaller numbers. So we can use that to our
advantage because we know that there's going to be a multiple of 13 in our
answer like it might be 13 times 3, or 13 times 7, or you know, something like that;
there's gonna be some multiple of 13 is gonna be one of the factors. And
that sort of restricts what numbers we can use because the multiples of 13 are
kind of... sparse; there's a lot of space between them. So let's just look at what
are the factors- the... the multiples of 13. So we've got 13 times 2 is 26, 13 times 3
is 39, 13 times 4 is 52. So what I'm saying is like, it might be something
like 25 26 27 28 - that might be the answer, or 38 39 40 41, something like that.
And that's going to help us find the answer because there aren't that many different
multiples of 13 that we'll need to try. So let's just write all these down...
that makes 65. 13 times 6 is 78. So we could just sort of work through some of
these, and see what happens. What I'll do first is just check that it's not
something like 13 times 14 times 15 times 16 or something like that.
See if we can make that. So 15 is 3 times 5 so I can get a 5 by pulling apart this
10, and I could get a 3 from there, so we could make 3 times 5 like that, and then
we could have 16 there. I can't make 17 though because that's a prime so we're
gonna have to go back and make 12 again. So I'll pull apart that.
12. And then we've got no option but to have 11 there, and then we've got this 3
left over. So we can't do it that way; it's gonna be involving a bigger
multiple of 13. Let's try the 13 times 2, so I might grab a 2 from there. Now,
can we make 25? 25 is actually 5 times 5 and we've only got the one 5 in here; we
actually need two 5's to make 25, so we can't do that. What about 27? You
might remember that from your 9 times tables or 3 times tables. That's 3 times 9.
So I could get a 3 from there, go 3 times 9. How about 28? So that's 4 times 7. I
can make that. But what about 29? Can you think of any factors of 29? We can tell
it's not divisible by 2, because it's odd. It's not divisible by 3, because what you
can do to check that is add up the digits, so 2 plus 9 is 11, and that's not
a multiple of 3, which means that that isn't a multiple of 3. Okay so 2 doesn't go
into it; 3 doesn't go into it. We don't need to check 4 because we've already
checked 2, and if 2 doesn't go into it then 4 won't. So how about 5? A number has to
end in 0 or 5 for it to be multiple of 5, so 5 doesn't go into it. We don't need to
check 6 because we've already checked 2, so... And 7 doesn't go into it. So 29 is
actually prime. So we have no way of making 29, and we've got no way of making
25. So that's not going to work; we can't make... we can't make that product with 26
so we can rule that out. All right let's move on to 13 by 3.
So I'm gonna pull that apart again and make that. Now we could think about maybe
having 40 there, so I can take the 10 and the 4 and combine those. What about 41?
That's odd so it's not divisible by 2. 4 plus 1 is 5 so that's not divisible by 3. It's
not divisible by 5 because it doesn't end in a 0 or 5. It's not in the 7 times
tables. So that one's actually prime again, and we can't make that out of any
of the numbers we have there so we can't do 41.
How about 38? 38 is divisible by 2 because it's even, so we could do 2 into
38... goes 19. Now can we make 19? 19 is actually prime as well. It's not divisible
by 2, by 3, by 5, nothing. So we actually have no way of making 38, and no
way of making 41. So we can rule out that as a possibility.
How about 13 times 4? Let's try that out. So I can make a 4 from there again and
then we could try making 53. So it's not even; 5 plus 3 is 8 so it's not divisible
by 3; it's not a multiple of 5; it's not in the 7 times tables; I think you'll
find that 53 is actually prime. So we can't make 53. What about 51? 5 plus 1 is
6 so that is actually multiple of 3. Let's try that out; let's go 3 into 51...
goes 1 remainder 2 and then 7... 17. We have no way of making a 17 because that's
prime. So we can't make 51 and we can't make 53, so we can rule that
one out. How about 13 times 5? Let's see if we can do that. So I can get
a 5 from the 10, combine those. And then, how about 66? So that's 6 times 11;
we can get that by combining those. I don't know about 67, but let's try 64, so
I would think of that as 8 times 8, so we've got one 8, we just need another
8. So I can make that out of... I can get a 2 from there, combine those, so 8
times 8... whoops! Get in there. That's 64, and then I'm left
with 9 and 7, and I can combine those to make 63! So that's the answer: 63
times 64 times 65 times 66. Let me know in the comments if you have any
questions or if you worked it out a different way, let me know how you did it.
Next time in this series I'll be doing this question: we have a long candle which
burns for seven hours, and a short candle which burns for 10 hours, and after 4
hours of burning both candles are the same height. We need to work out what was
the short candle's height divided by the long candle's height. So stay tuned for that
one. If you enjoyed this video then give it a thumbs up and consider subscribing.