Understanding the Greatest Common Factor (GCF): Methods and Examples

What is the Greatest Common Factor?

The greatest common factor (GCF), also known as the greatest common divisor (GCD) or highest common factor (HCF), of a set of whole numbers is the largest positive whole number that divides evenly into all the numbers in the set without leaving a remainder. Think of it as the biggest number that all the numbers in your set can be divided by perfectly. For instance, if you have the numbers 18, 30, and 42, their GCF is 6, because 6 is the largest number that divides into 18, 30, and 42 without any remainder.

The concept of GCF is fundamental in mathematics and has practical applications in simplifying fractions, solving algebraic problems, and even in computer science. Understanding how to find the GCF is a valuable skill in mathematical literacy.

Exploring the Greatest Common Factor of Zero

It’s interesting to consider the greatest common factor when zero is involved. Any non-zero whole number multiplied by zero equals zero. This means every non-zero whole number is technically a factor of zero.

For any whole number k, we know that k × 0 = 0, and therefore 0 ÷ k = 0.

For example, 7 × 0 = 0, so 0 ÷ 7 = 0. In this case, both 7 and 0 are factors of 0.

Therefore, the GCF of any non-zero whole number and zero is the non-zero whole number itself. For example, GCF(5, 0) = 5, and more generally, GCF(k, 0) = k for any whole number k.

However, the GCF(0, 0) is considered undefined in mathematics, as there’s no single “greatest” factor in this case.

Methods to Calculate the Greatest Common Factor

There are several methods to find the GCF of two or more numbers. The most suitable method often depends on the size of the numbers and how many numbers you are working with. Let’s explore three common methods: Factoring, Prime Factorization, and Euclid’s Algorithm.

Method 1: Factoring – Listing Common Factors

One straightforward method to find the GCF is by listing all the factors of each number in the set. Factors are whole numbers that divide evenly into a number without any remainder. Once you have listed the factors for each number, you identify the common factors – those that appear in all the lists. The greatest among these common factors is the GCF.

Example: Find the GCF of 18 and 27

1. List the factors of 18: 1, 2, 3, 6, 9, 18
2. List the factors of 27: 1, 3, 9, 27
3. Identify the common factors: 1, 3, 9
4. The greatest common factor is: 9

Example: Find the GCF of 20, 50, and 120

1. List the factors of 20: 1, 2, 4, 5, 10, 20
2. List the factors of 50: 1, 2, 5, 10, 25, 50
3. List the factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
4. Identify the common factors (present in all three lists): 1, 2, 5, 10
5. The greatest common factor is: 10

This method is effective for smaller numbers, but it can become time-consuming as the numbers get larger and have many factors.

Method 2: Prime Factorization – Using Prime Factors

Prime factorization is another effective method, especially for larger numbers. Prime factors are prime numbers that multiply together to give the original number. To use this method, you first find the prime factorization of each number. Then, you identify the prime factors that are common to all the numbers. For each common prime factor, you take the lowest power that appears in any of the factorizations. Finally, you multiply these common prime factors together to get the GCF.

Example: Find the GCF of 18 and 27

1. Prime factorization of 18: 2 × 3 × 3 (or 2 × 3²)
2. Prime factorization of 27: 3 × 3 × 3 (or 3³)
3. Identify common prime factors: The common prime factor is 3.
4. Lowest power of common prime factors: The lowest power of 3 present in both factorizations is 3² (or 3 × 3).
5. GCF: 3 × 3 = 9

Example: Find the GCF of 20, 50, and 120

1. Prime factorization of 20: 2 × 2 × 5 (or 2² × 5)
2. Prime factorization of 50: 2 × 5 × 5 (or 2 × 5²)
3. Prime factorization of 120: 2 × 2 × 2 × 3 × 5 (or 2³ × 3 × 5)
4. Identify common prime factors: The common prime factors are 2 and 5.
5. Lowest power of common prime factors: The lowest power of 2 is 2¹ (or 2), and the lowest power of 5 is 5¹ (or 5).
6. GCF: 2 × 5 = 10

Prime factorization can be more efficient than listing factors, especially when dealing with larger numbers, as it breaks down numbers into smaller, more manageable components.

Method 3: Euclid’s Algorithm – An Efficient Approach

Euclid’s Algorithm is a highly efficient method for finding the GCF of two numbers, especially very large numbers. It avoids the need to find factors or prime factors and relies on a series of divisions. The algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. A more efficient version uses the remainder of division.

Steps for Euclid’s Algorithm:

1. Divide the larger number by the smaller number and find the remainder.
2. If the remainder is 0, the smaller number is the GCF.
3. If the remainder is not 0, replace the larger number with the smaller number and the smaller number with the remainder.
4. Repeat steps 1-3 until the remainder is 0. The last non-zero remainder is the GCF.

To find the GCF of more than two numbers, you can find the GCF of the first two numbers, and then find the GCF of that result and the next number, and so on. For example, GCF(x, y, z) = GCF(GCF(x, y), z).

Example: Find the GCF of 18 and 27

1. Divide 27 by 18: 27 = 18 × 1 + 9 (remainder is 9)
2. Replace 27 with 18 and 18 with 9. Now find GCF of 18 and 9.
3. Divide 18 by 9: 18 = 9 × 2 + 0 (remainder is 0)
4. The last non-zero remainder was 9. Therefore, the GCF of 18 and 27 is 9.

Example: Find the GCF of 20, 50, and 120

First, find the GCF of 120 and 50 using Euclid’s Algorithm:

1. Divide 120 by 50: 120 = 50 × 2 + 20 (remainder is 20)
2. Divide 50 by 20: 50 = 20 × 2 + 10 (remainder is 10)
3. Divide 20 by 10: 20 = 10 × 2 + 0 (remainder is 0)
4. The GCF of 120 and 50 is 10.

Now, find the GCF of the result (10) and the third number (20):

1. Divide 20 by 10: 20 = 10 × 2 + 0 (remainder is 0)
2. The GCF of 20 and 10 is 10.

Therefore, the GCF of 20, 50, and 120 is 10.

Example: Find the GCF of 182664, 154875, and 137688

First, find the GCF of 182664 and 154875:

1. 182664 = 154875 × 1 + 27789
2. 154875 = 27789 × 5 + 15930
3. 27789 = 15930 × 1 + 11859
4. 15930 = 11859 × 1 + 4071
5. 11859 = 4071 × 2 + 3717
6. 4071 = 3717 × 1 + 354
7. 3717 = 354 × 10 + 177
8. 354 = 177 × 2 + 0

The GCF of 182664 and 154875 is 177.

Now, find the GCF of 177 and 137688:

1. 137688 = 177 × 777 + 159
2. 177 = 159 × 1 + 18
3. 159 = 18 × 8 + 15
4. 18 = 15 × 1 + 3
5. 15 = 3 × 5 + 0

The GCF of 177 and 137688 is 3.

Therefore, the GCF of 182664, 154875, and 137688 is 3.

Euclid’s Algorithm is particularly valuable when working with large numbers where factoring or prime factorization would be extremely tedious and time-consuming. It provides a systematic and efficient way to find the GCF.

Conclusion

Understanding the greatest common factor and knowing how to calculate it using different methods is a crucial skill in mathematics. Whether you choose factoring, prime factorization, or Euclid’s algorithm, each method provides a way to find the GCF, depending on the context and the numbers involved. From simplifying fractions to more complex mathematical problems, the concept of GCF plays a vital role in mathematical operations and problem-solving.

References

[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition. New York, NY: CRC Press, 2003 p. 101.
[2] Weisstein, Eric W. “Greatest Common Divisor.” From MathWorld–A Wolfram Web Resource.
Finding the Greatest Common Factor. Help With Fractions.
Euclidean Algorithm. Wikipedia.