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GCF Calculator + Online Solver With Free Steps
The GCF Calculator is an online application that helps in calculating the Greatest Common Factor for provided integers. The Greatest Common Factor is the factor with the highest common denominator among all factors involving two or more numbers.
The greatest common factor for any set of given numbers can be determined using either the listing approach or the prime factorization methodology.
What Is a GCF Calculator?
The GCF Calculator finds the biggest integer factor that exists between a set of numbers.
It is also referred to as the Highest Common Factor (HCF), the Greatest Common Denominator (GCD), or the Highest Common Divisor (HCD).
This is crucial in several mathematical applications, such as the simplification of polynomials, where it is frequently necessary to identify common components.
How To Use a GCF Calculator?
You can use the GCF Calculator by following the given detailed stepwise solution to find the required results. Simply follow the instructions to find the greatest common factor for the given data points.
Step 1
Input the given data points in the boxes specified on the calculator.
Step 2
Now press the “Submit” button to calculate the greatest common factor of the given data points, and also the whole step-by-step solution for the midpoint calculation will be displayed.
How Does GCF Calculator Work?
The GCF Calculator works by dividing the integer by its Greatest Common Factor, with the residue always equal to zero. The HCF or GCF (Greatest Common Factor) is another name for the GCD (Greatest Common Divisor) (Highest Common Factor).
The steps to determine the GCF of two or more numbers using the listing or factorization approach are provided below.
Each given number’s factors should be noted down.
- From the list of factors collected, make a list of all the common factors.
- The GCF of the given numbers will be given to us by the common factor with the highest value.
Several techniques can be used to locate GCF. While some of them are simple, others are more intricate. Knowing all will help you decide the one suitable:
- Using the list of factors,
- Prime factorization of numbers,
- Euclidean algorithm,
- Binary algorithm technique,
- Using multiple properties of GCF (including Least Common Multiple, LCM).
GCF Finder – List of Factors
The process of identifying all of the provided numbers’ components is the primary way to estimate the Greatest Common Divisor.
The initial value is simply produced by multiplying the factors, which are just numbers. Generally speaking, they can be both positive and negative. For instance, 2 x 3 equals six just as (-2) x (-3) equals 6.
As you can see, the process becomes more time-consuming and error-prone as the number of components increases.
Euclidean Algorithm
The principle on which the Euclidean algorithm is based states that if k is the greatest common factor of the numbers ‘A’ and ‘B,’ then ‘k’ is also the greatest common factor of their difference, A-B.
By repeating this process, we will eventually arrive at 0. The final non-zero value is the Greatest Common Divisor as a result.
Binary Greatest Common Divisor Algorithm
The Binary algorithm, also known as Stein’s algorithm, is absolutely for you if you want mathematical operations that are less complex than those utilized in the Euclidean algorithm (such as modulo). You only need to compare, subtract, and divide by two.
Keep in mind these identities while you calculate the greatest common factor of two numbers:
- Gcd (A, 0) = A, the fact that every number is divided by zero and observation from the last step in the Euclidean algorithm – one of the numbers drops to 0; therefore, the result was the one before.
- If A and B are even, it regards that gcd(A, B) = 2 x gcd(A2, B2) because we know that 2 is a common factor.
- If any one of the numbers is even, let’s say that number is A, then gcd(A, B) = gcd(A2, B). In this instance, two is not regarded as a common divisor, so the reduction will be continued until both of the numbers A and B become odd.
- If both the given A and B are odd and A≥B, then gcd(A, B)=gcd((A−B)2s, B). Now combine both of the characteristics in a single step.
- The first one is derived from the Euclidean algorithm, working out the Greatest Common Divisor of the difference between both numbers and the smaller one.
- The difference between two given odd numbers comes out to be even, due to which it can be divided by 2. Therefore the even one can be reduced as mentioned in step 3.
Coprime Numbers
Prime numbers are defined as numbers with no common factors. It is correct to say that they have no common divisors even though their lone common factor is 1, which is why we omit it from the prime factorization.
It can also be stated that the numbers ‘A’ and ‘B’ are coprime if:
GCF(A,B) = 1
The fact that the list of common components is empty does not necessarily imply that either of them is a prime number.
Coprime numbers include the pairs 5 and 7, 35 and 48, and 23156 and 44613.
Greatest Common Denominator of More Than Two Numbers
List all of the contributing reasons for each number because we can simply choose the most important one.
However, when the quantity of figures rises, it becomes evident that it takes an increasing amount of time.
The disadvantage of the prime factorization approach is similar, but since we can arrange all of the primes, for example, in ascending order, we can introduce a method to conclude a little quicker than before.
Solved Examples
Let’s explore some examples to understand the working of the GCF Calculator better.
Example 1
a). Find the GCF of 18 and 27
b). Find the GCF of 20, 50 and 120
Solution
(a).
The factors of 18 are given as follows:
1, 2, 3, 6, 9, and 18
The factors of 27 are given as:
1, 3, 9, and 27
The common factors of 18 and 27 are:
1, 3, and 9.
Therefore the GCF of 18 and 27 is 9.
(b).
The factors of 20 are given as:
1, 2, 4, 5, 10, and 20
The factors of 50 are given as:
1, 2, 5, 10, 25, and 50
The factors of 120 are given as:
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120
include Common factors of 20, 50, and 120 are given as:
1, 2, 5, and 10.
We will include the factors common to all three numbers.
Hence GCFs of 20, 50, and 120 are 10.
Example 2
Find the GCF (20, 50, 120)
Solution
The prime factorization of 20:
2 x 2 x 5 = 20
The prime factorization of 50:
2 x 5 x 5 = 50
The prime factorization of 120 :
2 x 2 x2 x 3 x 5 = 20
The common prime factors are given below:
2, 5
Therefore, the greatest common factor of 20, 50, and 120 is 2 x 5 = 10
Example 3
Find the GCF of the following:
GCF(182664, 154875, and 137688)
GCF (GCF(182664, 154875), 137688)
Solution
First, we find the GCF (182664, 154875)
182664 – (154875 x 1) = 27789
154875 – (27789 x 5) = 15930
27789 – (15930 x 1) = 11859
15930 – (11859 x 1) = 4071
11859 – (4071 x 2) = 3717
4071 – (3717 x 1) = 354
3717 – (354 x 10) = 177
354 – (177 x 2) = 0
So, the greatest common factor between 182664 and 154875 is 177.
Now we find the GCF (177, 137688)
137688 – (177 x 777) = 159
177 – (159 x 1) = 18
159 – (18 x 8) = 15
18 – (15 x 1) = 3
15 – (3 x 5) = 0
So, the GCF of 177 and 137688 is 3.
Therefore, the GCF of 182664, 154875, and 137688 is 3.