Prove that an even number squared is an even number.

Prove that an odd number squared is an odd number.

Prove that the sum of two odd numbers is even.

Prove that the sum of two even numbers is even.

Prove that the sum of an even number with an odd number is an odd number.

Prove that the sum of two consecutive integers is odd.

Four Basic Proof Techniques Used in Mathematics

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- Theorem:
- The sum of two consecutive integers is odd.

- Direct proof
- Proof by contradiction
- Proof by induction
- Proof by contrapositive

Prove that the product of two odd numbers is an odd number.

Prove that if \(x\) is even, then \(5x - 3\) is odd.

Prove that \(x\) is odd if and only if \(5x + 4\) is odd.

Let \(m, n, p \in \mathbb{Z}\). Prove that if \(m + n\) and \(n + p\) are even, then \(m + p\) is even.