Odd Number multiplied by Odd Number is Odd

Theorem

In the words of Euclid:

If an odd number by multiplying an odd number make some number, the product will be odd.

Let $a$ be odd and $b$ be even.

Then by definition of odd number:

$\exists c \in \Z: a = 2 c + 1$

$\exists d \in \Z: b = 2 d + 1$

So:

$\ds a \times b$

$=$

$\ds \paren {2 c + 1} \times \paren {2 d + 1}$

$\ds $

$=$

$\ds 4 c d + 2 c + 2 d + 1$

$\ds $

$=$

$\ds 2 \paren {2 c d + c + d} + 1$

Hence the result by definition of odd number.

$\blacksquare$