Question

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

Short answer

By using the method of direct proof, this Statement is true: If \(a \mid b\) and \(a \mid c,\) then \(a \mid (b+c)\).

Step-by-step solution

Understanding the Problem

The given problem expresses three integers \(a, b, c \in \mathbb{Z}\). It is known that \(a \mid b\) and \(a \mid c\). This means, there exist integers y and z such that \(b = ay\) and \(c = az\). Now, the task is to demonstrate that \(a \mid (b+c)\).

Formulating the Direct Proof

Taking the two equations described in Step 1: \(b = ay\) and \(c = az\). Summing both these equations gives us \(b + c = ay + az = a(y+z)\). Here, \(y + z\) is an integer because the sum of two integers is always an integer. Let's define \(k = y+z\). Therefore, we have \(b + c = ak\).

Finalizing the Proof

According to our statement \(b + c = ak\), here 'a' is a divisor of \(b+c\). In other words, this implies \(a \mid (b+c)\). Thus, our statement is proved.