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Perfect numbers

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7.2
PAULA KELLY: So we’ve already seen a number of ways of generating sequences. Let’s just see one more.
12.7
MICHAEL ANDERSON: OK, so let’s start with our first number 10. That’s going to be the first term of this particular sequence. To get to the next term, I’m going to write all the factors of 10 out first. So the factors of 1 and 10. 1 multiplied by 10 gives us 10. 2 and 5.
31.8
PAULA KELLY: So you’ve paired factors to make sure you have them all?
34.4
MICHAEL ANDERSON: Yeah.
34.8
PAULA KELLY: OK.
35.2
MICHAEL ANDERSON: So I’ve written all the factors down, and then I’m going to sum them. I’m going to add them together, but I’m not going to include the number itself 10. So 1, add 2, add 5, that gives us eight, and that’s the next term in our sequence. The next link in our factor chain.
50.9
PAULA KELLY: OK, so we need now factors of 8. So we’ll stick with our pairs. They’ve had 1, we’ll go with 8. And then we start with 1 looking at factors, didn’t we. OK, so you’ve got 1. The next logical thing to try would be 2.
62.8
MICHAEL ANDERSON: Yeah.
63.5
PAULA KELLY: I know 2 can be paired with 4. Our next logical number will be 3. We know 3 isn’t a factor. 4 is a factor. We have already, so we know we’re done.
72.4
MICHAEL ANDERSON: We can stop there. So we’ll use this to generate the next term in the sequence. We’ll discount the 8. We won’t use that one. Now we’re going to add 1, add 2, add 4 together, and that will give us 7.
84.5
PAULA KELLY: OK, so factor of 7. Just 1 in 7.
88.3
MICHAEL ANDERSON: Yeah, so 7 is a prime number, so just 1 and 7 will discount the 7, and we’re left with just one. So the next term is going to be 1. And I’m going to stop there.
99.3
PAULA KELLY: Because it will looking a factors of 1, we know it’s just 1.
102.9
MICHAEL ANDERSON: Yeah.
103.5
PAULA KELLY: We know our chain comes from summing the factors apart from the number itself, so nothing there to add.
109.7
MICHAEL ANDERSON: Yeah, so that’s how you construct a factor chain. So we could investigate this further. There’s lots of nice questions to have a go at, so if you make a different factor chain with a different number, will it always end with one? Will that penultimate number always be a prime number? Do you have a chain that always decreases in value at each step? What’s the shortest chain you could make, and what’s the longest chain? Have a play around and see what you discover.
135.3
PAULA KELLY: So now you’ve had a chance to try some number chains of your own, let’s explore it a bit more detail something you may have already come across.
143.1
MICHAEL ANDERSON: OK, so if you started a factor chain with the number 6, you might find something unusual. So let’s write the factors of 6 down and see what we get. So we’ve got 1 and 6, 2 and 3. And they’re the only four factors of 6. So if we don’t include the 6, and we add a 1 and 2 and 3 together, we get–
166.1
PAULA KELLY: We come back to 6.
167.6
MICHAEL ANDERSON: So our factor chain self repeats. So we come back to 6, and then we’ll get 6 and 6, and keeps on going forever.
175.3
PAULA KELLY: So there is a special term for this?
177.3
MICHAEL ANDERSON: Yeah, so 6 is part of a number family called perfect numbers. So a perfect number is perfect if all of its factors, excluding itself, sum to that number. So 6 is a perfect number.
189.1
PAULA KELLY: OK so 6 is a perfect number. Are there any other numbers are perfect number then?
193.8
MICHAEL ANDERSON: Yeah. There’s two perfect numbers between 1 and 30, so the next perfect number is 28. So you may want to pause the video now to convince yourself that 28 is a perfect number.
209.4
OK, so what are the factors of 28?
212.1
PAULA KELLY: Well we always start with 1. And then 1 goes into 28, 28 times.
215.5
SPEAKER 2: OK, so I’m going to write my pairs next to each other.
218.8
PAULA KELLY: OK, next logical one would be 2. I going to go 1 goes into 28 14 time. 3 would be the next one, not a multiple of 3. If we try 4, we know goes isn’t 28 7 times.
231.4
MICHAEL ANDERSON: OK, yeah.
232.4
PAULA KELLY: We could try five, it wouldn’t work. Six wouldn’t work. Seven does work. We’ve come back to this, so we know we have more.
238.6
MICHAEL ANDERSON: OK, so if we cross out the number 28, the number itself and then add all these factors together, 1, add 2 is 3, add 14 is 17, 4 gets us to 21, and then 7, we’re back to 28 again. So 28 is our next perfect number.
There are professional mathematicians, particularly pure mathematicians, whose jobs is to explore mathematics, find connections and make sense of how mathematics works.
When mathematicians make discoveries often these discoveries have no practical use at that moment in time. However, at a later date, someone may find an application for these discoveries.
An example of this is the exploration of prime numbers. At the time, mathematicians finding ever larger prime numbers seemed pointless. It was much later that these prime numbers were put to use in cryptography.
This next activity asks you to explore the sum of the factors of a number in order to discover what a perfect number is. There is no practical application of perfect numbers but it is a good classroom activity requiring students to practise finding factors of numbers.

Task

Write out the factors of 8.
Add up the factors of 8, but do not include 8 itself.
You should have got 1+2+4 = 7. This is less than 8.
As the sum of its divisors is less than the number itself we call 8 a “deficient number”.
Now, write out the factors of 12.
Add up the factors of 12, but do not include 12 itself.
Is your result less than or greater than 12?
If the sum of a number’s divisors is greater than the number itself we call it an “abundant number”.
In this video, Michael and Paula demonstrate a classroom activity which makes sequences by calculating the sum of the factors of numbers. There are many opportunities to make sequences of your own and to discover what a perfect number is. This is an excellent activity to develop fluency as students are required to find the factors of many numbers in order to create their sequence.

Teaching resource

This resource on perfect numbers is a presentation which can be used in the classroom explaining how to generate the sequences in this step. The extension to the activity explores a method for finding possible further perfect numbers.

Problem worksheet

Now complete question 10 and 11 from this week’s worksheet.
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Maths Subject Knowledge: Understanding Numbers

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