**Proving Two Things Are Equal**

If asked to do this in an exam it might be as simple as rearranging one side of an equation to show that it is the same as the other.**Example**: Prove (2w+2)² ≡ 4(w² + 2w + 1)

Expand the brackets on the left and then simplify them to show they equal to the right hand side.

(2w + 2)(2w +2) ≡ 4w² + 8w + 4 ≡ 4(w² + 2w + 1)

**NOTE:** notice the equals sign had 3 lines. This means that two things are equal for all values, not just certain values like in a normal equation.

Often though you are asked to prove that something is odd or even. For this you should remember the following:

- Any even number can be written 2n (anything multiplied by 2 is even)
- Any odd number can be written 2n + 1 therefore
- Consecutive numbers can be written and n, n+1, n+2 etc
- Consecutive even numbers can be written 2n, 2n + 2, 2n + 4 etc

**Example**: *w* is an even number. Explain why (*w* – 1)(*w* + 1) will always be odd.*Expand the brackets*

w² + w – w -1 = w² – 1

Any even number multiplied by itself is even. Therefore when you take one away from any value of w squared you will be left with an odd number. **Disproving By Example**

You might be asked to disprove something. You just have to find one value where the statement doesn’t hold up.**Example**: Liz says that when *m* > 1, *m*^{2} + 2 is never a multiple of 3.

Give a counter example to show that she is wrong.*Keep plugging in numbers until you find something an example that doesn’t work.*

Let m = 5 then *m*2 + 2 = 27, which is a multiple of 3!

So Liz was talking rubbish.