Prove that multiplication of two odd numbers produces an odd number.

An even number is any number that can be divided by two to give a whole number. If we define "n" as "any whole number", we can say that "2n" is an even number. Odd numbers cannot divide by 2 to give a whole number. We can say that "2n + 1" is an odd number. Using these definitions, we can write out the sum (2n + 1) x (2n + 1) = 4n^2 + 4n + 1, which represents the multiplication of two odd numbers. We must now transform this equation into some variation of our definition of an odd number. Factorising the first 2 terms gives us 2(2n^2 + 2n) + 1. The first term of 2(2n^2 + 2n) must be even, as regardless of what is inside the bracket, it is being multiplied by 2, which as we defined earlier, makes it even. This means we have some even number plus 1 (2n + 1), which again, as we defined earlier, makes an odd number. Therefore, any odd number multiplied by any other odd number gives an odd number.