write a convincing argument that the product of any two positive?

Question

odd integers is an odd integer

Answer 1

This would be a lot like the arguement that I gave you a little while ago, for the sum.
Only now you'd represent the product as (2a+1)(2b+1) =
2ab + 2a + 2b + 1 = 2(ab + a + b) + 1
Are you taking a course in number theory? You'll need to come up with all sorts of proofs for ideas like this. You'll start off with a "convincing arguement" like the ones we've given you here, then you'd firm it up into a formal proof.

Answer 2

2 is not a factor of any positive odd integer. Therefore, the product of two positive odd integers will also not have 2 as a factor. Every even integer must have 2 as a factor, so the product of two positive odd integers cannot be even and must be odd. QED

Answer 3

Odd integers can be defined as
odd = 2k + 1 (where k is an integer)
even = 2k (where k is an integer)

odd * odd = (2k + 1) * (2k + 1) = 4k^2 + 4k + 1

Hmm.... I'll have to come back to this later, I can't figure this out.

---- EDIT ----

Ahhh She-Nerd shows me the light... I tried showing that squaring (rather than multiplying two arbitrary odd numbers) an odd number is odd, and ran into trouble... oops.