Odd Numbers Multiplication Proof

Prove that if $x$ and $y$ are odd, then $xy$ is odd.

When tackling proofs in number theory, one fundamental concept is proving properties about parity, which is whether a number is odd or even. In this problem, you are asked to prove that the product of two odd numbers is odd. To approach such a proof, it's important to understand the definition of odd and even numbers. An integer is odd if it can be expressed in the form 2k + 1, where k is an integer. This representation is crucial because it allows you to manipulate the expressions algebraically based on this underlying concept.

In attempting this proof, a direct approach involves starting with the assumption that both numbers involved in the multiplication, x and y, are odd. Accordingly, you can express x as 2a + 1 and y as 2b + 1, where a and b are integers. The next step involves multiplying these expressions out and simplifying them to observe whether the outcome maintains the structure of an odd number, which again is checkable by seeing if it reduces to the form 2m + 1 for some integer m.

This task not only solidifies your understanding of basic number expressions but also enhances skills in logical reasoning and algebraic manipulation. Through such exercises, you become adept at constructing and deconstructing mathematical arguments, a vital skill in discrete mathematics and beyond.