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# What does the greatest common factor mean?

The greatest common factor (or GCF; sometimes called highest common factor) of two whole numbers is the largest whole number that is a factor of both. Take, for example, the numbers 12 and 15: The factors of 12 are 1, 2, 3, 4, 6, and 12; the factors of 15 are 1, 3, 5, and 15. Therefore, the common factors—or numbers in both lists of factors—are 1 and 3; therefore, the greatest (highest) common factor is 3.

There is another method used to discover the GCF: listing the numbers’ prime factors, then multiplying those numbers. For example, the prime factorization of 12 and 15 are: 2 × 2 × 3 = 12 and 3 × 5 = 15. Notice that the prime numbers have 3 in common; thus, the GCF is 3.

An example with larger numbers is to find the GCF of 36 and 54. Working it out by the first method, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36; the factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. The greatest (or highest) common factor of both numbers is 18. To work it out using prime factorization, the prime factorization of 36 is 2 × 2 × 3 × 3; the prime factorization of 54 is 2 × 3 × 3 × 3. Both these factorizations have one 2 and two 3s in common; thus, we multiply those common numbers, or 2 × 3 × 3 = 18.

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