What is proof and how does it work?

'Proving' something mathematically means showing that it is true for all possible cases. Therefore it isn't sufficient to simply give a lot of examples of a formula or test working; you need to find a way to show that it works for an infinite number of cases. GCSE proofs are pretty tricky compared with other topics on the course, so don't be too worried if you don't get it straight away, there's a certain way of thinking about numbers that you will pick up over time.
The aim of many proofs at GCSE is to use the question to come up with a formula or identity which can be used to show that a statement is true for all values. The best way to start this is to 'plug in' a few numbers to show yourself how the question works and to get used to what it is asking. Then, taking a generic number n, apply the same process to n as you did for the other numbers. Often, this will lead to an identity that proves the statement in the question. You will have to look for certain properties in this identity depending on the question. For example, if asked to prove that a certain expression of any number 'n' will be divisible by 2, you will be looking for a generic integer multiplied by 2 in your answer.