Firstly, we define a factor of a given number to be anything that that number is divisible by. For example, 1, 2, 3, and 6 are all the factors of 6.
Furthermore, we define a prime number as any number greater than 1 that has only two factors: itself and 1. So, 5 is a prime number, since its only factors are 1 and 5. However, the number 6, as mentioned, is divisible by 2 and 3 as well as 1 and 6, so it is not prime. We call the number 6 a composite number.
Note: it would be an impossible task to learn all the prime numbers off by heart – there are an infinity of them – so it’s important to understand what they are so to be able to spot one. That said, it is useful to become extra familiar with the first few, say: 2, 3, 5, 7, 11, 13, 17, as these are likely to appear a lot.
It follows that we define a prime factor of any given number to be any factor that number has that is also a prime number. Consider the number 10, for example. 10 is divisible by 1, 2, 5, and 10. Clearly 10 is not prime (it is divisible by 5 and 2), and 1 is also not prime (only numbers greater than 1 can be prime), whereas both 2 and 5 are prime. Thus, 2 and 5 are the prime factors of 10.
We can extend this idea further to talk about prime factorisations. Every positive whole number has a prime factorisation – a list of prime numbers that, when multiplied together, give you the original number. Furthermore, this list is unique for every number! Hence, they’re often called unique prime factorisations. In the example above, 10 has only two prime factors, 2 and 5, and the resulting prime factorisation of 10 is 2\times 5. This is a nice example, but it gets less obvious when the numbers get bigger and are divisible by the same prime factor multiple times. In more complicated cases, we use something called a factor tree to find a number’s unique prime factorisation.