# Video: GCSE Mathematics Foundation Tier Pack 2 • Paper 2 • Question 14

GCSE Mathematics Foundation Tier Pack 2 • Paper 2 • Question 14

03:59

### Video Transcript

The variables 𝑎 and 𝑏 are odd numbers, and 𝑐 is an even number. Part a) Give a suitable example to demonstrate that four multiplied by 𝑎 plus 𝑏 minus 𝑐 is a multiple of eight. Part b) show that four multiplied by 𝑎 plus 𝑏 minus 𝑐 is always a multiple of eight when 𝑎 and 𝑏 are odd numbers and 𝑐 is an even number.

The key here is that actually the result in the bracket that four is multiplied by must be even. This is because four multiplied by any even number is gonna be a multiple of eight. And that’s because if we had four multiplied by two for instance, you get eight which is a multiple of eight. And then for instance if we had four multiplied by four, that would give us 16 which again is a multiple of eight. Because actually, all we can do is actually think about the number 16. Four goes into it four times. Or eight will go into it half as many times. So eight goes into it twice, and so on.

So then, what you can do is actually find the numbers that fit, using trial and error. So I’ve chosen one set of values. And I’m gonna show you another set as well. So we’ve got 𝑎 is equal to one, because that’s an odd number. 𝑏 is equal to three which is also an odd number. And 𝑐 is equal to two which is an even number. Which is gonna give four multiplied by one plus three minus two, which is gonna give us four multiplied by two. And that’s because one plus three is four, take away two is two. Well, four multiplied by two is eight. And then eight multiplied by one is eight. So therefore, it’s actually a multiple of eight.

As I said, I’d also give you another example. So we’ve got 𝑎 is three, 𝑏 is 11, and 𝑐 is two. So just to use slightly bigger numbers this time. So therefore, this time we’d have four multiplied by three plus 11 minus two. Which gives us four multiplied by 12, because three add 11 is 14 take away two is 12. Well, this is equal to 48. Well, this can be rewritten as eight multiplied by six, so therefore is a multiple of eight. So, great. What we’ve done is given a couple of suitable examples to demonstrate that four multiplied by 𝑎 plus 𝑏 minus 𝑐 is a multiple of eight, when 𝑎 and 𝑏 are odd numbers and 𝑐 is an even number.

So in part a, we’ve actually given some examples to demonstrate this. However, in part 𝑏, what it wants us to do is to actually show that four multiplied by 𝑎 plus 𝑏 minus 𝑐 is always a multiple of eight, when 𝑎 and 𝑏 are odd numbers and 𝑐 is an even number. So I’ve actually already touched upon this when I talked about four multiplied by an even number is always gonna be a multiple of eight. But let’s dive into it a little deeper.

So if we think about our 𝑎 and our 𝑏, if we have an odd number add an odd number, it’s always going to be an even number. So that’s like the 𝑎 add the 𝑏 parts of our bracket because we’ve got an odd add an odd. And we also know that an even number minus an even number is gonna give us an even number. So it’s gonna be equal to an even number. And this is like our 𝑎 plus 𝑏 which is an even number, cause we’ve already ascertain that, minus 𝑐 because 𝑐 is an even number.

Well therefore, we can say that 𝑎 plus 𝑏 minus 𝑐 is even. And an even is always a multiple of two. So we know that even number is always a multiple of two. Because the even numbers are actually the two times table. So therefore, we can say that an even number is always two multiplied by an integer. So we’re gonna write that as two 𝑛.

So therefore, we can surmise that four multiplied by an even is always a multiple of eight. And that’s because four multiplied by an even is the same as four multiplied by two 𝑛. Well, four multiplied by two 𝑛 gives us eight 𝑛, which must be a multiple of eight because it’s eight multiplied by an integer.