prove that any odd number squared is one more than a multiple of four.

any odd number can be written as (2n+1), where n is any integer (whole number). Squaring any odd number is therefore= (2n+1)² . expanding the brackets gives =4n²+2n+2n+1² = 4n²+4n+1. factorising the 4 out of the first two terms gives =4(n²+n)+1. 4(n²+n) is a multiple of 4 due to the factored out 4, and the +1 after means that any odd number squared is one more than a multiple of 4.