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.