## gcf, hcf, gcd (1,260; 28) = ?

### Approach 1. Integer numbers divisibility:

#### Divide the larger number by the smaller one.

#### Notice that dividing our numbers leaves no remainder:

#### 1,260 ÷ 28 = 45 + 0;

#### => 1,260 = 28 × 45;

#### So, 1,260 is divisible by 28;

#### 28 is a factor (a divisor) of 1,260;

#### Consequently, greatest (highest) common factor (divisor):

gcf, hcf, gcd (1,260; 28) = 28

## gcf, hcf, gcd (1,260; 28) = 28 = 2^{2} × 7;

1,260 is divisible by 28

### Approach 2. Integer numbers prime factorization:

#### Prime Factorization of a number: finding the prime numbers that multiply together to make that number.

#### 1,260 = 2^{2} × 3^{2} × 5 × 7;

1,260 is not a prime, is a composite number;

#### 28 = 2^{2} × 7;

28 is not a prime, is a composite number;

** Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself. *

* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.

### Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd:

#### Multiply all the common prime factors, by the lowest exponents (if any).

#### gcf, hcf, gcd (1,260; 28) = 2^{2} × 7

## gcf, hcf, gcd (1,260; 28) = 2^{2} × 7 = 28;

1,260 has all the prime factors of the number 28.

## Final answer:

Greatest (highest) common factor (divisor)

gcf, hcf, gcd (1,260; 28) = 28 = 2^{2} × 7;

1,260 is divisible by 28.

1,260 has all the prime factors of the number 28.

### Why do we need the greatest (highest) common factor (divisor)?

#### When you have calculated the greatest (highest) common factor (divisor), GCF (HCF, GCD), of the numerator and denominator of a fraction, it becomes easier to reduce it (simplify it) to the lowest terms.

### More operations of this kind:

## Calculator: greatest common factor (divisor) gcf, gcd

## 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

#### If two numbers "a" and "b" have no other common factors (denominators) than one, gfc, gcd, hcf (a; b) = 1, then the numbers "a" and "b" are called COPRIME, or prime to each other.

#### If "a" and "b" are not coprime, then every common factor of "a" and "b" is a also a factor (divisor) of the greatest common factor, GCF (greatest common divisor, GCD, highest common factor, HCF) of "a" and "b".