Prove that the square of an odd number is always 1 more than a multiple of 4

This is a nice little question off the 2018 Edexcel Higher Maths paper! When we heard the word 'prove' in a question at this level, automatically we should be thinking about using algebra to help us out! Firstly, we need to remember that even numbers and odd numbers can be written in the form 2n and 2n+1 respectively, with n being any integer. This means that if we want to write out the form of "the square of an odd number", we could write it out as (2n+1)² Expanding the brackets, this is written as 4n²+4n+1. Looking back at the question, we are trying to prove that our number is "always 1 more than a multiple of 4". Looking at our formula, we can factorise the first two terms as follows: 4n²+4n+1 = 4(n² +n) +1, which gives us a number that is one more than a multiple of 4