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Please provide integers separated by a comma "," and click the "Calculate" button to find their common factors.
What is a factor?
A factor is a term in multiplication. For example, in:
3 × 4 = 12,
3 and 4 are the factors. It is possible for a number to have multiple factors. Using 12 as an example, in addition to 3 and 4 being factors:
3 × 4 = 12
2 × 6 = 12
1 × 12 = 12
It can be seen that 1, 2, 3, 4, 6, and 12 are all factors of the number 12. This is the most basic form of a factor, but algebraic expressions can also be factored, though that is not the intent of this calculator.
What is a common factor?
A common factor is a factor that is shared between two different numbers. It can also be referred to as a common divisor. As an example:
The factors of 16 include: 1, 2, 4, 8, and 16.
The factors of 12 include: 1, 2, 3, 4, 6, and 12.
Thus, the common factors of 16 and 12 are: 1, 2, and 4.
Often in math problems, it can be desirable to find the greatest common factor of some given numbers. In this case, the greatest common factor is 4.
This calculator only accepts positive integers as input to calculate their common factors. While only two numbers are used in the above example, the calculator can compute the common factors of more than two numbers.
Please provide numbers separated by a comma "," and click the "Calculate" button to find the GCF.
What is the Greatest Common Factor (GCF)?
In mathematics, the greatest common factor (GCF), also known as the greatest common divisor, of two (or more) non-zero integers a and b, is the largest positive integer by which both integers can be divided. It is commonly denoted as GCF(a, b). For example, GCF(32, 256) = 32.
Prime Factorization Method
There are multiple ways to find the greatest common factor of given integers. One of these involves computing the prime factorizations of each integer, determining which factors they have in common, and multiplying these factors to find the GCD. Refer to the example below.
| EX: | GCF(16, 88, 104) 16 = 2 × 2 × 2 × 2 88 = 2 × 2 × 2 × 11 104 = 2 × 2 × 2 × 13 GCF(16, 88, 104) = 2 × 2 × 2 = 8 |
Prime factorization is only efficient for smaller integer values. Larger values would make the prime factorization of each and the determination of the common factors, far more tedious.
Euclidean Algorithm
Another method used to determine the GCF involves using the Euclidean algorithm. This method is a far more efficient method than the use of prime factorization. The Euclidean algorithm uses a division algorithm combined with the observation that the GCD of two integers can also divide their difference. The algorithm is as follows:
GCF(a, a) = a
GCF(a, b) = GCF(a-b, b), when a > b
GCF(a, b) = GCF(a, b-a), when b > a
In practice:
- Given two positive integers, a and b, where a is larger than b, subtract the smaller number b from the larger number a, to arrive at the result c.
- Continue subtracting b from a until the result c is smaller than b.
- Use b as the new large number, and subtract the final result c, repeating the same process as in Step 2 until the remainder is 0.
- Once the remainder is 0, the GCF is the remainder from the step preceding the zero result.
| EX: | GCF(268442, 178296) 268442 - 178296 = 90146 178296 - 90146 = 88150 90146 - 88150 = 1996 88150 - 1996 × 44 = 326 1996 - 326 × 6 = 40 326 - 40 × 8 = 6 6 - 4 = 2 4 - 2 × 2 = 0 |
From the example above, it can be seen that GCF(268442, 178296) = 2. If more integers were present, the same process would be performed to find the GCF of the subsequent integer and the GCF of the previous two integers. Referring to the previous example, if instead the desired value were GCF(268442, 178296, 66888), after having found that GCF(268442, 178296) is 2, the next step would be to calculate GCF(66888, 2). In this particular case, it is clear that the GCF would also be 2, yielding the result of GCF(268442, 178296, 66888) = 2.