### Video Transcript

The variables π and π are odd
numbers, and π is an even number. Part a) Give a suitable example to
demonstrate that four multiplied by π plus π minus π is a multiple of eight. Part b) show that four multiplied
by π plus π minus π is always a multiple of eight when π and π are odd numbers
and π 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 thatβs 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 Iβve chosen one set of
values. And Iβm gonna show you another set
as well. So weβve got π is equal to one,
because thatβs an odd number. π is equal to three which is also
an odd number. And π 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 thatβs 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, itβs actually a
multiple of eight.

As I said, Iβd also give you
another example. So weβve got π is three, π is 11,
and π is two. So just to use slightly bigger
numbers this time. So therefore, this time weβd 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 weβve done is given a couple
of suitable examples to demonstrate that four multiplied by π plus π minus π is a
multiple of eight, when π and π are odd numbers and π is an even number.

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

So if we think about our π and our
π, if we have an odd number add an odd number, itβs always going to be an even
number. So thatβs like the π add the π
parts of our bracket because weβve 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 itβs gonna be equal to an even
number. And this is like our π plus π
which is an even number, cause weβve already ascertain that, minus π because π is
an even number.

Well therefore, we can say that π
plus π minus π 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 weβre gonna write that as two
π.

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