A visual proof for product of two odd numbers being odd?

Question

Is there a visual proof showing that product of two odd numbers is odd? Or product of a number and an even number is always even?

I've got some idea for addition and subtraction.

X X X X X + X X X X X X X

= [X X] [X X] {X} + [X X] [X X] [X X] {X}

= [X X] [X X] [X X] [X X] [X X] + {X X}

Visually, odd numbers cannot be separated into pairs, without something being left over.

But if we add two odd numbers, the 'left out' Xs for both odd numbers will form a pair, and hence the sum will be even.

Answer 1

The product of two odd numbers drawn on a square grid is a rectangle with one square in the middle and everything else symmetric, so even. Even plus one is odd.

In algebra,

$$(2a+1)(2b+1)=1+2(a+b)+4ab,$$

where $1$ is the center square, $2(a+b)$ are horizontal and vertical strips around the center square and $4ab$ are the remaining four corners.

For instance, the product $7·5$ is depicted below, with A the square, B the strips and C the corners.

CCCBCCC

CCCBCCC

BBBABBB

CCCBCCC

CCCBCCC

The product of two even numbers does not have the center square or strips, it has only corners.

$$2a·2b=4ab$$

Answer 2

This is coming from the Pierre de Fermat, where each odd number has representation as the difference of two squares. If $a$ and $b$ are odd numbers then their product ($N$) is always odd.

It is worth notice here the fact that $N$ can be represented as an arithmetic progression of odd numbers, and the number of terms is always odd since smaller side of rectangle ($a$) is odd. If the number of terms in the sequence would be even, then $N$ becomes an even number.

You can also see that the square $T$ cannot be odd if $N+T$ is odd.

Answer 3

If $m$ and $n$ are odd an $(m\times n)$-array $A$ of lattice points has a center dot $c$. Reflection of $A$ in $c$ produces orbits of length $2$ and the special orbit $\{c\}$. It follows that $|A|$ is odd.

Answer 4

A square with odd side can be partitioned as follows:

Conclusion: $Odd\cdot Odd = Even + Even + Odd = Odd.$