6- Use a direct proof to show that the product of two odd numbers is odd

“A direct proof of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true. A direct proof shows that a conditional statement p → q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs.” Discrete Mathematics and its Applications by Rosen.

Let n and m be two odd numbers.

n=2k+1 and p=2t+1 for some integers k and t (definition of odd number).


np=2k2t + 2k + 2t + 1

np= 2(k2t + k + t) + 1 (definition of odd number).

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