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Finding the Greatest Common Factor (GCF) of Two or More Expressions
To find the greatest common factor (GCF) of two or more expressions, follow a four-step procedure built on prime factorization:
- Factor each coefficient into primes. Write all variables with exponents in expanded form. Break each numerical coefficient into its prime factors, and expand every variable power so each individual factor is visible (e.g., write as ).
- List all factors — matching common factors in a column. In each column, circle the common factors. Align the prime factors and expanded variable factors vertically so that identical factors line up; then circle the factors that appear in every expression.
- Bring down the common factors that all expressions share. Collect one copy of each circled factor.
- Multiply the factors. The product of the collected common factors is the GCF.
This process is analogous to the prime factors method for finding the LCM, except that the GCF uses only the factors shared by all expressions rather than all factors from any expression.
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