3.4: The Greatest Common Factor (2024)

Definition: Common Factor

We notice that 2 appears as a factor in both numbers, that is, 2 is a common factor of 30 and 42. We also notice that 3 appears as a factor in both numbers. Three is also a common factor of 30 and 42.

Definition: Greatest Common Factor (GCF)

When considering two or more numbers, it is often useful to know if there is a largest common factor of the numbers, and if so, what that number is. The largest common factor of two or more whole numbers is called the greatest common factor, and is abbreviated by GCF. The greatest common factor of a collection of whole numbers is useful in working with fractions (which we will do in [link]).

How to find the greatest common factor (GCF) of two or more whole numbers

1. Write the prime factorization of each number, using exponents on repeated factors.
2. Write each base that is common to each of the numbers.
3. To each base listed in step 2, attach the smallest exponent that appears on it in either of the prime factorizations.
4. The GCF is the product of the numbers found in step 3.

Sample Set A

Find the GCF of the following numbers.

12 and 18

Solution

1. \(\begin{array} {l} {12 = 2 \cdot 6 = 2 \cdot 2 \cdot 3 = 2^2 \cdot 3} \\ {18 = 2 \cdot 9 = 2 \cdot 3 \cdot 3 = 2 \cdot 3^2} \end{array}\)
2. The common bases are 2 and 3.
3. The smallest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 1 and 1 (\(2^1\) and \(3^1\)), or 2 and 3.
4. The GCF is the product of these numbers.
   \(2 \cdot 3 = 6\)

The GCF of 30 and 42 is 6 because 6 is the largest number that divides both 30 and 42 without a remainder.

Sample Set A

18, 60, and 72

Solution

1. \(\begin{array} {l} {18 = 2 \cdot 9 = 2 \cdot 3 \cdot 3 = 2 \cdot 3^2} \\ {60 = 2 \cdot 30 = 2 \cdot 2 \cdot 15 = 2 \cdot 2 \cdot 3 \cdot 5 = 2^2 \cdot 3 \cdot 5} \\ {72 = 2 \cdot 36 = 2 \cdot 2 \cdot 18 = 2 \cdot 2 \cdot 2 \cdot 9 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 = 2^3 \cdot 3^2} \end{array}\)
2. The common bases are 2 and 3.
3. The smallest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 1 and 1:
   \(2^1\) from 18
   \(3^1\) from 60
4. The GCF is the product of these numbers.
   GCF is \(2 \cdot 3 = 6\)

Thus, 6 is the largest number that divides 18, 60, and 72 without a remainder.

Sample Set A

700, 1,880, and 6,160

Solution

1. \(\begin{array} {rcl} {700 \ = \ 2 \cdot 350 \ = \ 2 \cdot 2 \cdot 175} & = & {2 \cdot 2 \cdot 5 \cdot 35} \\ {} & = & {2 \cdot 2 \cdot 5 \cdot 5 \cdot 7} \\ {} & = & {2^2 \cdot 5^2 \cdot 7} \\ {1,880 \ = \ 2 \cdot 940 \ = \ 2 \cdot 2 \cdot 470} & = & {2 \cdot 2 \cdot 2 \cdot 235} \\ {} & = & {2 \cdot 2 \cdot 2 \cdot 5 \cdot 47} \\ {} & = & {2^3 \cdot 5 \cdot 47} \\ {6,160 \ = \ 2 \cdot 3,080 \ = \ 2 \cdot 2 \cdot 1,540} & = & {2 \cdot 2 \cdot 2 \cdot 770} \\ {} & = & {2 \cdot 2 \cdot 2 \cdot 2 \cdot 385} \\ {} & = & {2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 77} \\ {} & = & {2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 7 \cdot 11} \\ {} & = & {2^4 \cdot 5 \cdot 7 \cdot 11} \end{array}\)
2. The common bases are 2 and 5
3. The smallest exponents appearing on 2 and 5 in the prime factorizations are, respectively, 2 and 1.
   \(2^2\) from 700.
   \(5^1\) from either 1,880 or 6,160.
4. The GCF is the product of these numbers.
   GCF is \(2^2 \cdot 5 = 4 \cdot 5 = 20\)

Thus, 20 is the largest number that divides 700, 1,880, and 6,160 without a remainder.