### Video Transcript

π, π, and π are positive whole numbers. π is a prime number. ππ minus ππ is a prime number. Show that π and π are consecutive numbers.

First, letβs just be clear on some of the wording in the question. First of all, prime numbers are numbers which have exactly two factors. We use this definition rather than the description that they have factors of one and themselves because this creates some confusion about whether or not one is a prime number. One has factors of one and itself because itself is one. But it doesnβt have exactly two factors. It has only one factor. So one is not a prime number. If we use the definition of a prime number as having exactly two factors, then this doesnβt create any confusion.

However, it is still true that if a number is prime, then its only factors will be one and itself. So if π is a prime number, then its only factors will be π and one. Equally, if ππ minus ππ is a prime number, then its only factors will be itself, ππ minus ππ, and one. Now, weβre asked to show that π and π are consecutive numbers. That just means integers that come one after the other, such as six and seven. And to do so, letβs look at this number ππ minus ππ in more detail.

We can see that the two terms that weβre subtracting do in fact have a common factor of π. We can therefore factorize the expression ππ minus ππ as π multiplied by π minus π. But this would mean that the expression ππ minus ππ has factors π and π minus π as well as one and itself. This canβt be the case if ππ minus ππ is a prime number. So it must be the case then that π and π minus π are actually equal to one and the original number itself. Thatβs ππ minus ππ.

But π, remember, is prime. So π canβt be equal to one as one isnβt a prime number for the reasons we discussed earlier. Therefore, it must be the case that π is actually equal to ππ minus ππ. And the other factor, π minus π, is equal to one. But if π minus π is equal to one, then adding π to each side of this equation, we see that π is equal to π plus one. But if π is equal to π plus one, then this means that π is the next whole number up after π, which tells us that π and π are consecutive. This is what we originally asked to show.

So by factorizing ππ minus ππ and then using the fact that both π and ππ minus ππ were prime, we show that in fact the number π is equal to ππ minus ππ. And π minus π is equal to one, which means that π is equal to π plus one, making π and π consecutive numbers.