Greatest Common Factor Calculator
| Enter Numbers ▼ |
|---|
| Number 1= | |
| Number 2= | |
| Number 3= | |
| Number 4= |
| * - add up to 7 numbers |
Cite
Share
Print
Result |
GCF =
4
(Greatest Common Factor)
The GCF calculation using Euclidean algorithm
▶ Sort down the numbers, heading from the lowest number upwards (in ascending order). So, the new superset will be: 36, 72, 128, 284
▶ Get the first pair of numbers setting the smaller one 36 as a divisor. Conduct sequential division of the previous denominator and the remainder (R) of its division until we've got a zero in the remainder.
▶ Finding GCF of 72 and 36:
72 ÷ 36 = 2 (R0)
GCF(72, 36) = 36 is the sub-result, as we have more number pairs.
▶ Finding GCF of 128 and 36:
128 ÷ 36 = 3 (R20)
36 ÷ 20 = 1 (R16)
20 ÷ 16 = 1 (R4)
16 ÷ 4 = 4 (R0)
GCF(128, 36) = 4 is the sub-result, as we have more number pairs.
▶ Finding GCF of 284 and 4:
284 ÷ 4 = 71 (R0)
GCF(284, 4) = 4 provides with the final answer.
GCF(36, 72, 128, 284) = 4
What is the Greatest Common Factor?
The Greatest Common Factor (GCF), co-called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF), represents the largest positive integer that divides multiple numbers without a remainder. Understanding how to find the GCF is fundamental. Whether you're simplifying fractions, solving equations, or busy with the other mathematical tasks, understanding of GCF is absolutely necessary.
\[ \text{For numbers } a, b: \]
\[ \text{GCF}(a,b) = \max{d \in \mathbb{Z}^+ : d | a \text{ and } d | b} \]
Ways of Finding the Greatest Common Factor (GCF)
Several methods can be used to calculate the GCF, including listing factors, prime factorization, and the Euclidean algorithm. Each method for finding the GCF has its advantages depending on the context. For the instance of small numbers, listing factors or prime factorization may suffice, while the Euclidean Algorithm excels with larger numbers due to its efficiency and simplicity.
1. Listing Factors Method
One of the most straightforward ways to find the GCF is to list the factors of each number and identify the largest common factor. For example, to find the GCF of 18 and 24:
- Factors of 18: \(1, 2, 3, 6, 9, 18\)
- Factors of 24: \(1, 2, 3, 4, 6, 8, 12, 24\)
- Common factors: \(1, 2, 3, 6\)
GCF(18, 24) = 6.
This method works well for small numbers but becomes impractical for larger ones.
2. Prime Factorization Method
This method is efficient for numbers with clear prime factors. Useful in educational settings. Prime factorization involves breaking each number into its prime factors and then finding the common factors. To find the GCF of \(48\) and \(60\):
- Prime factors of \(48 = 2^4 \times 3\)
- Prime factors of \(60 = 2^2 \times 3 \times 5\)
- Common factors: \(2^2 \times 3\)
GCF (48, 60) = 12.
3. Euclidean Algorithm
The Euclidean Algorithm is a highly efficient and systematic method for finding the GCF. And it is especially powerful for very large numbers and forms the basis of many modern computational approaches. It works by repeatedly subtracting or dividing the smaller number into the larger number until the remainder is zero.
Mathematical representation:
\[ \text{GCF}(a,b) = \text{GCF}(b, a \bmod b) \]
For example, to find the GCF of \(56\) and \(98\):
1. Divide \(98\) by \(56\): remainder \(42\).
2. Divide \(56\) by \(42\): remainder \(14\).
3. Divide \(42\) by \(14\): remainder \(0\).
GCF(56, 98) = 14.
Sources
Rosen, Kenneth H. Elementary Number Theory and Its Applications. Boston, Addison-Wesley, 2011., p.45-78.
Hardy, G.H., Wright, E.M., Silverman, J. and Wiles, A. (2008). An Introduction to the Theory of Numbers. Oxford University Press., p.102-135.
Knuth, D.E. (1998). The Art of Computer Programming. Addison-Wesley Professional., p.333-345.
