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Hi, I'm Miss Davies.

In this lesson, we're going to be looking at proof by counter example.

Mo says, if you square a number and then add one, the result is always prime.

We've been asked to give a counter example to show that Mo is incorrect.

Let's start by squaring one and adding one.

This gives a result of two.

What about if we square two and add one? That gives a result of five.

What about three? That gives a result of 10, and 10 is not a prime number.

If we try this with five, we get a result of 26, which is also not a prime number.

With this, we have given two counter examples to show that Mo is incorrect.

Whitney says that prime numbers are always odd.

We're going to give a counter example to show that Whitney is incorrect.

A prime number has exactly two factors.

Two is the first prime number, then three, five, seven, 11, 13, 17, 19, and so on.

These numbers are odd but two is even.

This is the counter example to show that Whitney is incorrect.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

The difference of two even numbers is even, but the product of two odd numbers is not even.

My example is that three multiplied by seven is 21.

But you might have multiplied two different odd numbers together.

Here is a question for you to try.

Pause the video to complete your task and resume once you're finished.

Here is the answer.

Nine is a non-prime odd number that is between two and 14.

We've been asked to prove that the statement that n squared subtract n add 11 is prime for all positive integer values of n.

We are going to prove that this is false.

Let us start by working out the result if n is between one and five.

When n is one, this gives a result of 11 and 11 is prime.

When n is two, this gives a result of 13, which is also prime.

When n is three, this gives the result of 17, which is again, prime.

When n is four, the result is 23, which is prime.

And when n is five, the result is 31, which again is prime.

I've then worked out the value of n squared subtract n add 11 for the values of n between six and 10.

All of these gave a prime result.

If we look at 11, this gives a result of 121, which is not prime.

As 11 is a factor of it as well as one and itself.

When n is equal to 11, n squared subtract n add 11 is equal to 121.

121 is not prime as it has more than two factors.

This is the counter example.

We are going to prove that this following statement is false.

k squared is greater than or equal to k for all values of k.

If we say that k is an integer between one and five, let's find the values of k squared.

If k is one, then k squared is also one.

These are equal to each other so the statement is true.

When k is two, k squared is four.

This is greater than two.

If k is three, then k squared is nine, which is greater than the value of k.

When k is four, k squared is 16, which again is true for the rule.

When k is five, k squared is 25, which fits this statement.

Let's try some special cases.

If k isn't an integer, for example, 2.

5.

This means that k squared is 6.

25.

This is greater than the value of k.

When k is negative, for example, negative three, this gives that k squared is nine, which again, fits the statement.

When k is between zero and one, for example, 0.

5.

This means that k squared is 0.

25, which is less than the value of k.

This statement is false when k is equal to 0.

5 Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

For question four, if a is equal to zero then 4a is equal to 5a.

You might've looked at negatives for this question as these would also show that the statement is false.

Here are some questions for you to try, pause the video, to complete your task and resume once you're finished.

Here are the answers.

You might have found a different counter example to use, which is great.

That's all for this lesson.

Thanks for watching.