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Hi, I'm Miss Davies.

In this lesson, we're going to be looking at number proofs, focusing on odd and even numbers.

How can we represent odd and even numbers algebraically? If n is an integer, this is greater than or equal to zero.

An even number can be represented as 2n.

It could also be represented by multiplying any expression by two.

An odd number is represented by 2n + 1 or by multiply any expression by two and adding one.

In our first example, we're going to prove that the product of two even numbers is even.

If we multiply 2n by 2n + 2, this is product of two consecutive even numbers.

We are looking at the product of two even numbers, not specifically consecutive ones.

To do this, we're going to multiply 2n by 2m as these represent any two even numbers.

This gives the result of 4nm.

As we can take two out as a factor, it shows us that this is a multiple of two and is therefore even.

In our next example, we're going to prove that the product of any two odd numbers is odd.

If we were to multiply 2n + 1 by 2n + 3, this would give us the product of two consecutive odd numbers.

We want to find the product of any two odd numbers.

Therefore, we're going to say that m and n are any integers that are greater than or equal to zero.

Our first odd number is represented by 2n + 1.

Our second odd number is represented by 2m + 1.

This gives us the product of any two odd numbers.

If we expand this bracket, it gives us 4mn + 2n + 2m + 1.

We can take two out as a factor of the first three terms. This leaves us with two multiplied by our expression add one.

This represents an odd number.

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, x + x + 1 simplifies to 2x + 1 and odd numbers are always one more than even numbers.

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.

Make sure that you're subtracting all terms within this bracket to give your answer.

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, two multiplied by any expression is always an even number.

One more than this is always an odd number.

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, one more than two lots of an expression is always representing an odd number.

That's all for this lesson.

Thanks for watching.