2026-10-29

Exercises for Chapter 7

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Chapter 7” (pp. 155–156).

A. Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters 4–7.

• 1. Suppose \(x\in\mathbb{Z}\). Then \(x\) is even if and only if \(3x+5\) is odd.

• 6. Suppose \(x,y\in\mathbb{R}\). Then \(x^3+x^2y=y^2+xy\) if and only if \(y=x^2\) or \(y=-x\).

• 7. Suppose \(x,y\in\mathbb{R}\). Then \((x+y)^2=x^2+y^2\) if and only if \(x=0\) or \(y=0\).

• 12. There exists a positive real number \(x\) for which \(x^2<\sqrt{x}\).

• 15. Suppose \(a,b\in\mathbb{Z}\). Prove that \(a+b\) is even if and only if \(a\) and \(b\) have the same parity.

• 18. There is a set \(X\) for which \(\mathbb{N}\in X\) and \(\mathbb{N}\subseteq X\).

• 20. There exists an \(n\in\mathbb{N}\) for which \(11\mid(2^n-1)\).

• 24. If \(a\in\mathbb{Z}\), then \(4\nmid(a^2-3)\).

• 31. If \(n\in\mathbb{Z}\), then \(\gcd(n,n+1)=1\).

• 35. Suppose \(a,b\in\mathbb{N}\). Then \(a=\gcd(a,b)\) if and only if \(a\mid b\).