Question

Use the method of direct proof to prove the following statements. Suppose \(a, b \in \mathbb{Z}\). If \(a \mid b\), then \(a^{2}\) | \(b^{2}\).

Short answer

Based on the definitions and direct proof method applied, the statement that if \(a \mid b\), then \(a^{2}\) divides \(b^{2}\), is proven to be true.

Step-by-step solution

Understanding the Definitions

Start by refreshing the definitions necessary for this problem. An integer \(a\) divides another integer \(b\), written as \(a \mid b\), if there exists another integer \(k\) such that \(b = a \cdot k\). Squaring on both sides, we get \(b^{2} = a^{2} \cdot k^{2}\). Our goal is to prove that \(a^{2}\) divides \(b^{2}\).

Apply the Definitions to the Problem

Given that \(a \mid b\), we know that there exists an integer \(k\) such that \(b = a \cdot k\). Applying this knowledge, square both sides to get \(b^{2} = a^{2} \cdot k^{2}\).

Deduce the Final Result

Notice that the right-hand side of the equation in step 2, \(a^{2} \cdot k^{2}\), indicates that \(a^{2}\) divides \(b^{2}\), because it can be written in the form of \(a^{2} \cdot l\) where \(l = k^{2}\) and \(k^{2}\) is an integer as the square of any integer is another integer.