## Tutoring: what is it and how to calculate the greatest common factor GCF of integers numbers (also called greatest common divisor GCD, or highest common factor, HCF)

#### If "t" is a factor (divisor) of "a" then among the prime factors of the prime factorization of "t" will appear only prime factors that also appear in the prime factorization of "a", and the maximum of their exponents is at most equal to those involved in the prime factorization of "a".

For example, 12 is a divisor of 60:

- 12 = 2 × 2 × 3 = 2
^{2}× 3 - 60 = 2 × 2 × 3 × 5 = 2
^{2}× 3 × 5

#### If "t" is a common factor of "a" and "b", then the prime factorization of "t" contains only prime factors involved in the prime factorizations of both "a" and "b", by the lower powers (exponents).

For example, 12 is the common factor of 48 and 360.

- 12 = 2
^{2}× 3 - 48 = 2
^{4}× 3 - 360 = 2
^{3}× 3^{2}× 5 - Please note that 48 and 360 have more factors (divisors): 2, 3, 4, 6, 8, 12, 24. Among them, 24 is the greatest common factor, GCF (or the greatest common divisor, GCD, or the highest common factor, HCF) of 48 and 360.

#### The greatest common factor, GCF, is the product of all the prime factors involved in both the prime factorizations of "a" and "b", by the lowest powers.

Based on this rule it is calculated the greatest common factor, GCF, (or greatest common divisor GCD, HCF) of several numbers, as shown in the example below:

- 1,260 = 2
^{2}× 3^{2} - 3,024 = 2
^{4}× 3^{2}× 7 - 5,544 = 2
^{3}× 3^{2}× 7 × 11 - Common prime factors are: 2, its lowest power is min. (2; 3; 4) = 2; 3, its lowest power is min. (2; 2; 2) = 2;
- GCF (1,260; 3,024; 5,544) = 2
^{2}× 3^{2}= 252