Question 10 of 12, Step 1 of 1 9/12 Correct Find the GCF (greatest common factor) of the following terms. \[ \{27 x y, 81,54 x y\} \]

To find the GCF of the terms \(27xy\), \(81\), and \(54xy\), we start by determining the prime factorizations of each term. For \(27xy\), its prime factorization is \(3^3 \cdot x \cdot y\). For \(81\), the factorization is \(3^4\). For \(54xy\), we have \(2 \cdot 3^3 \cdot x \cdot y\). The GCF will include the lowest power of each prime factor present in all terms. Thus, for the prime factor \(3\), the minimum power is \(3^3\). The GCF also includes the factors \(x\) and \(y\) since they appear in the terms \(27xy\) and \(54xy\). Therefore, the GCF is: \[ GCF = 3^3 \cdot x \cdot y = 27xy. \] Now you have the largest common factor among your terms—easy peasy! Keep in mind that finding the GCF is a great way to simplify fractions and solve equations. Know that the concept of GCF can be traced back to ancient mathematicians, including the Greeks, who utilized it in their algorithmic problem-solving approaches. The Euclidean algorithm, which can efficiently find the GCF of two numbers, dates back to Euclid's "Elements" in around 300 BC!