Prove the product of two odd numbers is always odd

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.

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