<p>If \(A = \{(x,y) : xy = 8,\; x,y \in \mathbb{Z}\}\), then \(n(A) =\)</p>
Step-by-Step Solution
Key Concept: List all integer divisor pairs of 8: (1,8),(2,4),(4,2),(8,1),(-1,-8),(-2,-4),(-4,-2),(-8,-1). That is 8 ordered pairs.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Integer factor pairs of 8 (positive): $(1,8),(2,4),(4,2),(8,1)$ — 4 pairs. Negative pairs: $(-1,-8),(-2,-4),(-4,-2),(-8,-1)$ — 4 more. Total $n(A)=8$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: 2