2026-10-22

Exercises for Chapter 6

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Chapter 6” (p. 144).

A. Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.)

• 2. Suppose \(n\in\mathbb{Z}\). If \(n^2\) is odd, then \(n\) is odd.

• 3. Prove that \(\sqrt[3]{2}\) is irrational.

• 6. If \(a,b\in\mathbb{Z}\), then \(a^2-4b-2\neq 0\).

• 10. There exist no integers \(a\) and \(b\) for which \(21a+30b=1\).

• 11. There exist no integers \(a\) and \(b\) for which \(18a+6b=1\).

• 14. If \(A\) and \(B\) are sets, then \(A\cap(B-A)=\emptyset\).

• 15. If \(b\in\mathbb{Z}\) and \(b\nmid k\) for every \(k\in\mathbb{N}\), then \(b=0\).

• 16. If \(a\) and \(b\) are positive real numbers, then \(a+b\geq 2\sqrt{ab}\).

B. Prove the following statements using any method from Chapters 4, 5 or 6.

• 19. The product of any five consecutive integers is divisible by 120. (For example, the product of 3, 4, 5, 6 and 7 is 2520, and \(2520=120\cdot 21\).)

• 20. We say that a point \(P=(x,y)\) in \(\mathbb{R}^2\) is rational if both \(x\) and \(y\) are rational. More precisely, \(P\) is rational if \(P=(x,y)\in\mathbb{Q}^2\). An equation \(F(x,y)=0\) is said to have a rational point if there exist \(x_0,y_0\in\mathbb{Q}\) such that \(F(x_0,y_0)=0\). For example, the curve \(x^2+y^2-1=0\) has rational point \((x_0,y_0)=(1,0)\). Show that the curve \(x^2+y^2-3=0\) has no rational points.

• 24. The number \(\log_2 3\) is irrational.