Suppose and are two odd numbers, then   and for some . We have

where .

An integer is an odd number if it is two times another integer plus 1. Therefore the product of two odd numbers is  four times the product of two other integers  plus two  times the sum of the two other integers plus one. This shows that the product of two odd numbers is odd

Suppose n and m are two odd numbers, then m=2k+1 and n=2l+1 for some k, l are in Z. We have

m n=(2k+1)(2l+1)=4kl+2k+2l+1=2(2kl+k+l)+1=2q+1

where q=2kl+k+l.

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Suppose and are two odd numbers, then   and for some . We have

where .