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.

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.