What is the Greatest Common Factor (GCF) of 27cd and 72cde?

Understanding the greatest common factor (GCF) is essential in both algebra and number theory. The GCF of two or more numbers or expressions is the largest expression that divides each of them without leaving a remainder. In this article, we will explore the GCF of the algebraic expressions 27cd and 72cde. By the end, you will be able to identify the common prime factors and calculate the GCF step by step.

Prime Factorization of 27cd and 72cde

To determine the GCF, we first need to break down each expression into its prime factors.

Prime Factors of 27cd

The prime factorization of 27cd is as follows:

Prime factors of 27cd  3 × 3 × 3 × cd  33cd

Prime Factors of 72cde

The prime factorization of 72cde is:

Prime factors of 72cde  2 × 2 × 2 × 3 × 3 × 3 × cd  23 × 33cd

Identifying Common Prime Factors

The next step involves identifying the common prime factors between 27cd and 72cde. By comparing the prime factorizations, we can see that both expressions contain the term 33cd.

Common Prime Factors

The common prime factors between 27cd and 72cde are 33cd.

Calculating the GCF

The greatest common factor (GCF) is the product of the common prime factors. Therefore, we multiply the common prime factors together:

GCF  33cd  27cd

Thus, the GCF of 27cd and 72cde is 27cd.

Conclusion and Further Understanding

By breaking down the expressions into their prime factors, we can easily identify and multiply the common factors to find the GCF. In this case, the GCF of 27cd and 72cde is 27cd. Understanding how to calculate the GCF is crucial in solving many algebraic equations and simplifying expressions.

To further your understanding, consider practicing with other algebraic expressions and exploring the relationship between the GCF and other algebraic concepts. For more resources and a deeper dive into similar problems, refer to additional lessons on prime factorization and GCF in algebra.