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

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.

• Multiplying means that you have to arrange the Xs in a rectangular shape. Now, Pick two odd numbers, and construct the rectangle of those sizes: then try to couple the Xs and see what you get. – Crostul May 29 '17 at 7:23
• Makes some sense. Thanks. – yomayne May 29 '17 at 7:24
• On a (2m+1) by (2n+1) rectangle, working from the lower left to upper right, draw some horizontal and vertical lines to divide it into 2x2 squares, There will remain an upper row divided into 1x2 rectangles and a right column of 2x1 rectagles, and a single 1x1 square at the top right: Odd x odd = odd. – DanielWainfleet May 29 '17 at 17:22

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.

$$2a·2b=4ab$$

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.

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.

A square with odd side can be partitioned as follows:

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