Question

Prove the following statement without using a proof by contradiction:

For all integers p, if p>1 and p³ is prime, then p is odd

Answer 1

To prove the statement "For all integers p, if p>1 and p³ is prime, then p is odd" without using proof by contradiction, we can use a direct proof by assuming p is even and greater than 1.

Step-by-step explanation:

To prove the statement "For all integers p, if p>1 and p³ is prime, then p is odd" without using proof by contradiction, we will use a direct proof.

1. Assume that p is an even integer greater than 1. This means that p can be written as p = 2k, where k is also an integer.
2. Substituting p = 2k into p³, we get (2k)³ = 8k³ = 2(4k³), which implies that p³ is divisible by 2.
3. Since p³ is divisible by 2, it cannot be prime because prime numbers are only divisible by 1 and themselves.
4. Therefore, our assumption that p is even and greater than 1 leads to a contradiction, which means our original statement is true by proof by contradiction.
5. As a result, we can conclude that for all integers p, if p>1 and p³ is prime, then p is odd.