“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=(2k+1)(2t+1)

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

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

⧠

**Related exercises:**

- 3. Show that the square of an even number is an even number using a direct proof
- 4. Show that the additive inverse, or negative, of an even number is an even number using a direct proof
- 5. Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p is even. What kind of proof did you use?
- 7. Use a direct proof to show that every odd integer is the difference of two squares.