Lesson video

Hello, and welcome to another lesson.

My name is Mr. Maseko.

Before you get on with today's lesson, make sure you have a pen and something to write on.

So pause the video here and go get those things.

Okay, now that you have those things, in today's lessons we'll be talking about factors and primes.

So let's get on with it.

Try this activity: Fill in the blanks in the calculations below using positive integers.

Pause the video and give this a go.

Okay.

Let's see what you've come up with.

Well, 8 is equal to blank times blank.

What could we have used? 4 times 2, good.

We could have put 4 times 2.

Is there anything else I could have said? Well, we could have said 8 times 1.

Is there anything else we could have used? No, because those are the only positive integers.

Remember, integers are whole numbers.

So these are the only positive integers that multiply together to give you 8.

What about 16? Well, 16, we could say 4 times 4.

What else? Well, we could say 8 times 2.

What else? We could say 16 times 1.

There anything else? How do you know you found all the different combination of integers that multiply to give you this numbers? The other pairs that you could have had, you could have wrote 1 time 16, which is the same as writing 16 multiplied by 1.

Did you draw diagrams to represent your images? Well, if we think of factor pairs: now, a definition for you to write down: So a factor of a number is an integer.

What did we say integers were? The whole number.

So it's an integer that divides exactly into another number without leaving a remainder.

So without a remainder.

So if you look at 8, so 8 has 4 factors, which are 1, 2, 4, and 8.

And you can organise those factors into factor pairs so the 2 numbers that multiply together to give you 8 is 8 times 1, and we also have 4 times 2.

Can you see how these diagrams represent 8 times 1 and 4 times 2? Well, that first diagram is 8 lots of those single boxes, 8 times 1.

And that second diagram, it's 4 lot of 2's.

Is there another way we could have drawn these diagrams? Is there another way? Well, for the 4 lots of 2's, we could have gone 2 lots of 4's, if you just look at the diagram the other way.

That is 2 lots of 4's.

Let's look at 16.

How many factors does 16 have? And can you draw diagrams to represent all the factor it has.

So pause the video here and give that a go.

Okay, now that you've done this, let's see what you could have come up with.

If we look at 16, 16 has 5 distinct factors: So 1, 2, 4, 8, and 16.

Our factor pairs are 16 times 1; 8 times 2; and 4 times 4.

So if you look, 4 is a repeated, 4 is a repeated factor.

This pen is refusing to write.

Let's see.

So 4 is a repeated, there we go, factor.

We got there in the end.

So there are only 5 distinct factors because 4 is repeated in this factor pair.

Now, we'll talk more about numbers with repeated factors in a later lesson.

Pause the video here and give this Independent task a go.

Okay, now that you've tried this, let's see what you come up with.

So state all the factor numbers for the integers from 1 to 12.

What do you notice? Now, you should have noticed that some of the numbers only have 1 factor pair.

Those numbers are 1, 2, 3, 5, 7, and 11, but 1 only has 1 factor, which was 1.

The others have 2 factors.

So 2, 3, 5, 7, 11, all have 2 factors.

And those numbers are special numbers and write this down: 2, 3, 5, 7, 11 are what we call prime numbers.

And they're numbers with exactly 2 factors.

So why is 1 not a prime number.

Well, 1 only has 1 factor, which is 1.

While prime numbers have to have 2 factors, exactly 2 factors.

And the first 5 prime numbers are 2, 3, 5, 7, and 11.

Now, 12 had the most factors, and an interesting fact about 12, we call that an abundant number because the sum of its factors that are not itself, so 1, 2, 3, 4, and 6, is larger than 12.

And 1 has the least number of factors, which is just 1.

You should have also noticed that there were 3 numbers with an odd number of factors.

And those were 1, 4, and 9.

And that's because they have a repeated factor.

So 1 has 1 as its repeated factor.

4 has 2 as its repeated factor.

And 9 has 3 as its repeated factor.

And 1, 4, and 9 are the first 3 square numbers.

And we'll talk more about square numbers in a later lesson.

Here's an Explore task for you to try.

We've defined prime numbers.

Remember, a prime number has exactly 2 factors.

So find the first 10 prime numbers.

And how many even numbers from 4 to 30 can be written as the sum of 2 primes? Pause the video here and give this a go.

Hello everyone.

It's Miss Jones here.

Hello, Mr. Maseko.

I'm coming in here just to do the solution to this bit for you.

Find the first 10 prime numbers.

Remember our definition at the top.

Some people think 1 is a prime number, but that's a common misconception.

1 isn't a prime number because it doesn't have exactly 2 factors, it only has 1.

The first prime number, if you remember, is 2.

The next one then is 3, then 5, then 7, then 11, 13, 17, 19, 23, and 29.

And let's just check we've got 10 there.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Okay.

Let's move on to our next question.

How many of the even numbers from 4 to 30 can be written as the sum of 2 primes? And we had some examples there.

So I've put the answers up to that one in the table there.

And really interestingly, we could write all of these numbers as the sum of 2 primes.

And for some of them there's more than one way of doing it.

So have a look at the table and see if it matches some of your answers and see if there are any there that you didn't think of.

In actual fact, someone called Goldbach stated that for every even number we can express it as the sum of 2 primes.

And this theory, although is unproven, has been tested up to 400 trillion.

An interesting fact for you, you know, can anybody prove this conjecture to be wrong? I don't know, but I thought I'd leave you with that interesting fact.

Okay, let's give you back to your teacher who can finish off the lesson for you.

So thank you for participating in today's lesson.

I look forward to seeing you again next time.