GCF Calculator
A GCF calculator finds the greatest common factor of two or more whole numbers. The greatest common factor, also called the greatest common divisor, is the largest positive number that divides each given number without leaving a remainder.
What Is the GCF?
The GCF is the largest factor shared by two or more numbers. For example, the factors of 12 include 1, 2, 3, 4, 6, and 12, while the factors of 18 include 1, 2, 3, 6, 9, and 18.
The common factors are 1, 2, 3, and 6, so the GCF of 12 and 18 is 6.
How to Find the GCF
One simple method is to list the factors of each number and identify the largest factor they have in common.
For example, consider 24 and 36:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The largest common factor is 12, so:
GCF(24, 36) = 12
Using Prime Factorization
The GCF can also be found by breaking each number into its prime factors. For example:
24 = 2 Γ 2 Γ 2 Γ 3
36 = 2 Γ 2 Γ 3 Γ 3
The prime factors shared by both numbers are 2 Γ 2 Γ 3.
2 Γ 2 Γ 3 = 12
Therefore, the GCF is 12.
GCF Using the Euclidean Algorithm
For larger numbers, the Euclidean algorithm can be a quicker method. Divide the larger number by the smaller number and use the remainder in the next division. Continue until the remainder becomes zero. The last non-zero remainder is the GCF.
For example:
48 Γ· 18 = 2 remainder 12
18 Γ· 12 = 1 remainder 6
12 Γ· 6 = 2 remainder 0
Therefore, the GCF of 48 and 18 is 6.
GCF of More Than Two Numbers
The same idea can be applied to three or more numbers. Find the GCF of two numbers first, then find the GCF of that result and the next number.
For example:
GCF(12, 18) = 6
GCF(6, 30) = 6
Therefore, the GCF of 12, 18, and 30 is 6.
GCF vs. LCM
The GCF and least common multiple (LCM) solve different types of problems. The GCF is the largest number that divides the given numbers evenly, while the LCM is the smallest positive number that is a multiple of all the given numbers.
GCF is often useful when simplifying fractions or dividing items into equal groups. LCM is commonly used when working with repeating schedules or finding a common denominator.
Common Uses of the GCF
The greatest common factor is useful for simplifying fractions, factoring algebraic expressions, dividing objects into equal groups, and solving problems involving divisibility.
Common Mistakes to Avoid
Make sure you are looking for a factor, not a multiple. The GCF must divide every given number exactly. Also remember that the greatest common factor is the largest shared positive factor, not simply any common factor.
Quick Summary
The GCF is the largest positive whole number that divides two or more numbers without a remainder. It can be found by listing factors, using prime factorization, or applying the Euclidean algorithm. A GCF calculator provides a quick way to find and verify the result.