2026-10-08

Exercises for Chapter 4

Source: Richard Hammack, The Book of Proof, 3rd ed., “Exercises for Chapter 4” (pp. 126–127).

A. Use the method of direct proof to prove the following statements.

• 1. If \(x\) is an even integer, then \(x^2\) is even.

• 6. Suppose \(a,b,c\in\mathbb{Z}\). If \(a\mid b\) and \(a\mid c\), then \(a\mid(b+c)\).

• 11. Suppose \(a,b,c,d\in\mathbb{Z}\). If \(a\mid b\) and \(c\mid d\), then \(ac\mid bd\).

• 18. Suppose \(x\) and \(y\) are positive real numbers. If \(x<y\), then \(x^2<y^2\).

• 20. If \(a\) is an integer and \(a^2\mid a\), then \(a\in\{-1,0,1\}\).

• 26. Every odd integer is a difference of two squares. (Example: \(7=4^2-3^2\), etc.)