# Why is the product of 2 odd numbers, odd?

Mar 14, 2018

See an explanation below:

#### Explanation:

The same odd number added together will always produce and even number.

If $a$ is odd (or even for that matter) then

$a + a = 2 a$

Because 2 times a number is always even.

If $a$ and $b$ are odd numbers then we can write their product as:

$a \times b$

This can be rewritten as:

$a \times \left(b - 1 + 1\right) \implies$

$a \times \left(\left(b - 1\right) + 1\right) \implies$

$\left(a \times \left(b - 1\right)\right) + \left(a \times 1\right)$

Because $b$ is odd, therefore $\left(b - 1\right)$ is even.

Because $\left(b - 1\right)$ is even, therefore $a \times \left(b - 1\right)$ is even.

Because $a$ is odd and $\left(a \times 1\right) = a$, therefore, $\left(a \times 1\right)$ is also add.

Therefore:

Because an odd number plus an even number equal and odd number, therefore:

$\left(a \times \left(b - 1\right)\right) + \left(a \times 1\right) = \text{even number" + "odd number} =$

$\text{odd number}$