GCF Calculator

What Is the Greatest Common Factor?

The Greatest Common Factor (GCF) is the largest positive integer that divides two or more numbers exactly. It is also called the Highest Common Factor (HCF) or Greatest Common Divisor (GCD).

For example, the factors shared by 12 and 18 are 1, 2, 3, and 6. Because 6 is the largest of these common factors, the GCF of 12 and 18 is 6.

How to Find the GCF

1. Write down the numbers you want to compare.
2. Find the factors or prime factorization of each number.
3. Identify the factors shared by all the numbers.
4. Choose the largest common factor.
5. Check that the GCF divides every input number exactly.

GCF Example: 48 and 72 Using the Euclidean Algorithm

Step 1: Divide the larger number, 72, by 48. The quotient is 1 and the remainder is 24.

Step 2: Now divide 48 by the previous remainder, 24. The quotient is 2 and the remainder is 0.

Step 3: When the remainder becomes 0, the last non-zero remainder is the GCF.

The result can be checked because 48 ÷ 24 = 2 and 72 ÷ 24 = 3, so 24 divides both numbers exactly.

GCF Using Prime Factorization

Prime factorization breaks a number into prime factors. To find the GCF, use the prime factors that appear in every number and take the smallest exponent of each shared prime.

Consider 36 and 60.

Step 1: The prime factors shared by both numbers are 2 and 3.

Step 2: Use the smallest exponent for each shared prime: 2² and 3¹.

Therefore, the GCF of 36 and 60 is 12.

Finding the GCF of Three Numbers

The GCF can also be found for three or more numbers. Consider 18, 30, and 42.

Step 1: Write the prime factorization of each number.

Step 2: The prime factors common to all three numbers are 2 and 3. Their smallest exponents are both 1.

Therefore, the GCF of 18, 30, and 42 is 6.

GCF of Algebraic Expressions

The same idea can be used when factoring algebraic expressions. For example, consider 12x² and 18x.

The numerical GCF of 12 and 18 is 6, while the smallest power of the shared variable x is x.

The expression can therefore be factored as:

Common Mistakes to Avoid

• Choosing a factor that is not common: The GCF must divide every number in the calculation exactly.
• Choosing a smaller common factor: Several factors may be shared, but the GCF is the largest one.
• Stopping the Euclidean algorithm too early: Continue dividing until the remainder becomes 0. The last non-zero remainder is the GCF.
• Using the largest exponents for the GCF: In prime factorization, use the smallest exponent of each prime shared by all numbers.
• Confusing GCF with LCM: The GCF is the largest common factor, while the LCM is the smallest positive common multiple.
• Ignoring variable powers in expressions: When factoring algebraic terms, use the smallest power of each variable common to all terms.

Why Use a GCF Calculator?

A GCF Calculator can quickly find the greatest common factor of two or more numbers and show the calculation steps. It is useful for checking manual work, simplifying problems, and finding common factors when the numbers are large or have several prime factors.