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Chapter 6 Proofs

Here are the definitions of even and odd.

  • An integer \(n \in \mathbb{Z}\) is even when there exists \(k \in \mathbb{Z}\) such that \(n = 2k\text{.}\)

  • An integer \(n \in \mathbb{Z}\) is odd when there exists \(k \in \mathbb{Z}\) such that \(n = 2k+1\text{.}\)

Here are the definitions of divisible and not divisisble. Let \(n,d \in \mathbb{Z}\) be integers.

  • We say that \(d \mid n\) when there exists \(k \in \mathbb{Z}\) such that \(n=dk\text{.}\)

  • Otherwise, we say that \(d \nmid n\text{.}\) This means that \(n \neq dk\) for all \(k \in \mathbb{Z}\text{.}\)

Use these definitions to prove the following statements.

Exercises Practice Problems

1. Odd to Even.

Prove that for all integers \(n \in \mathbb{Z}\text{,}\) if \(n\) is odd, then \(5n + 3 \) is even.

2. Any to Odd.

Prove that if \(n \in \mathbb{Z}\text{,}\) then \(5n^2+3n+7\) is odd.

3. Even/Odd Combo.

Prove that if \(a\) is any odd integer and \(b\) is any even integer, then \(2a+3b\) is an even integer.

4. Summing Squares.

Prove that if \(a\) is any odd integer and \(b\) is any even integer, then \(a^2+b^2\) is an odd integer.

5. Proof by Contrapositive.

Use a proof by contrapositive to show that for all integers \(m,n \in \mathbb{Z}\text{,}\) if \(mn\) is even then \(m\) is even or \(n\) is even.

6. Proof by Contradiction.

Use a proof by contradiction to show that for any integer \(n \in \mathbb{Z}\text{,}\) we have \(4 \nmid n^2-2.\)

7. Your Choice.

Prove the following statement using either proof by contrapositive, or proof by contradiction.

For integers \(a,b,n \in \mathbb{Z}\text{,}\) if \(ab = n\text{,}\) then either \(a \geq \sqrt{n}\) or \(b \geq \sqrt{n}\text{.}\)