Step 2: Determine the common factors. \( \begin{array}{l}27=3 \cdot 3 \cdot 3 \\ 45=3 \cdot 3 \cdot 5\end{array} \)

When looking for common factors, we can break down each number into its prime factors. For 27, which is \(3^3\), and 45, which is represented as \(3^2 \cdot 5\), we can see that both numbers share a factor of 3. Specifically, the greatest common factor (GCF) can be found by taking the lowest power of each prime factor common to both numbers, which in this case is \(3^2\), equating to 9. Finding common factors comes in handy outside of math class! For instance, when simplifying fractions, knowing the GCF helps reduce them to their simplest form, making calculations easier and clearer. It’s widely used in fields like engineering and computer science, where optimizing resources and energy can hinge on understanding these principles. So, the next time you're tackling a math problem, think of it as a real-world strategist simplifying the complexity!