# Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 3 • Question 15

𝜉 is the set of all integers from 1 to 𝑘. 𝐸 is the set of even numbers. 𝑚 is the set of multiples of 5. 𝑃 is the set of prime numbers. a) Explain why there is only one number in 𝐸 ∩ 𝑃. b) Given that 𝑘 is a multiple of 10, how many numbers are in 𝑚 ∩ 𝜉? Circle your answer. [A] (𝑘/5) − 1 [B] (𝑘/5) + 1 [C] 𝑘 − 5 [D] 𝑘/5

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### Video Transcript

𝜉 is the set of all integers from one to 𝑘. 𝐸 is the set of even numbers. 𝑚 is the set of multiples of five. 𝑃 is the set of prime numbers. Part a, explain why there is only one number in 𝐸 intersection 𝑃. Part b, given that 𝑘 is a multiple of 10, how many numbers are in 𝑚 intersection 𝜉? Circle your answer. Is it 𝑘 divided by five minus one, 𝑘 divided by five plus one, 𝑘 minus five, or 𝑘 divided by five?

Let’s just remind ourselves of some of the notation that we need to know. This is the Greek letter 𝜉. And it’s usually used to denote the universal set. That’s all the objects we’re interested in. And here, those are all integers. Remember, those are whole numbers from one to 𝑘. So we could say that that could be one, two, three, four, all the way through to 𝑘.

We then see that 𝐸 is the set of all even numbers. Remember, an even number is a multiple of two. And it will always end in two, four, six, eight, or zero. 𝑚 is the set of multiples of five. And those are the numbers in the five times tables. They’re five, 10, 15, 20, and so on. And then, we have 𝑃 as being the set of prime numbers. Remember, a prime number has exactly two factors, one and itself. The first five prime numbers are two, three, five, seven, and 11. It’s important to remember that one is not a prime number. One actually only has one factor, one. So how does this help us for part a?

Well, this symbol here that looks a little bit like the letter n means the intersection of two sets. When we think about that in terms of two circles on a Venn diagram, it’s the overlap. So we’re interested in the numbers that would appear in the overlap between the set of even numbers and the set of prime numbers. That is the set of numbers that are both even numbers and prime numbers. Two is the only even prime number. This means it is the only number which will appear in the intersection between sets 𝐸 and 𝑃.

Remember, we said a prime number has exactly two factors. Two has two factors; it’s one and two. Any other even number will have a factor of two apart from one and itself. So there’s no way that any other even number can have only two factors. And we can now see that there can only be one number in 𝐸 intersection 𝑃. It’s the number two, since two is the only even prime number.

Now, let’s consider part b. We’re told that 𝑘 is a multiple of 10. Remember, multiples of 10 are the numbers in the 10 times table: 10, 20, 30, and so on. And we’re looking to find how many numbers there are in the intersection of sets 𝑚 and 𝜉. That means we’re interested in the multiples of five which are also integers from one to 𝑘. If 𝑘 is equal to 10, then numbers that appear in this set are five and 10. If 𝑘 is equal to 20, the multiples of five in this set are five, 10, 15, and 20. And if 𝑘 is equal to 30, the multiples are five, 10, 15, 20, 25, and 30.

That means that when 𝑘 is equal to 10, there are two numbers in the set. When 𝑘 is equal to 20, there are four numbers in the set. And when 𝑘 is equal to 30, there are six numbers in the set. In fact, we can see that for every 10 numbers there are in 𝜉, there are two multiples of five. When 𝑘 is a multiple of 10, then we can say that two out of every 10 of these numbers are multiples of five. That’s two-tenths of 𝑘.

Remember, “of” is commonly interchanged with the multiplication symbol. And we can simplify two-tenths to one-fifth. And we say that one-fifth of 𝑘 are multiples of five when 𝑘 is a multiple of 10. Another way of saying this is 𝑘 over five. So our answer here is 𝑘 over five.