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.
Best Answer
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.
$$2a·2b=4ab$$