You might have learned that the product of two odd numbers is always odd, but do you how to prove it?

We can prove it by using a similar strategy we used in Why is the sum of two even numbers also an even number

we can represent any odd number with 2m+1, where m is any integer.

now lets say we have 2m+1 and 2n+1, two odd number.

we multiply them together:

(2m+1)*(2n+1) = 4mn+2n+2m +1

(4mn+2n+2m)+1

2(2mn+n+m)+1

2(2mn+n+m) is guaranteed to be an even number, and any even number plus one is an odd number.