## gcf, hcf, gcd (109; 144) = ?

### Approach 1. Integer numbers prime factorization:

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

#### 109 is a prime number, it cannot be broken down to other prime factors;

#### 144 = 2^{4} × 3^{2};

144 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).

#### But the two numbers have no common prime factors.

#### gcf, hcf, gcd (109; 144) = 1

## gcf, hcf, gcd (109; 144) = 1;

Coprime numbers (relatively prime).

### Approach 2. Euclid's algorithm:

#### This algorithm involves the operation of dividing and calculating remainders.

#### 'a' and 'b' are the two positive integers, 'a' >= 'b'.

#### Divide 'a' by 'b' and get the remainder, 'r'.

#### If 'r' = 0, STOP. 'b' = the GCF (HCF, GCD) of 'a' and 'b'.

#### Else: Replace ('a' by 'b') & ('b' by 'r'). Return to the division step above.

#### Step 1. Divide the larger number by the smaller one:

144 ÷ 109 = 1 + 35;

Step 2. Divide the smaller number by the above operation's remainder:

109 ÷ 35 = 3 + 4;

Step 3. Divide the remainder from the step 1 by the remainder from the step 2:

35 ÷ 4 = 8 + 3;

Step 4. Divide the remainder from the step 2 by the remainder from the step 3:

4 ÷ 3 = 1 + 1;

Step 5. Divide the remainder from the step 3 by the remainder from the step 4:

3 ÷ 1 = 3 + 0;

At this step, the remainder is zero, so we stop:

1 is the number we were looking for, the last remainder that is not zero.

This is the greatest common factor (divisor).

#### Greatest (highest) common factor (divisor):

gcf, hcf, gcd (109; 144) = 1

## gcf, hcf, gcd (109; 144) = 1;

coprime numbers (relatively prime).

## Final answer:

Greatest (highest) common factor (divisor)

gcf, hcf, gcd (109; 144) = 1;

Coprime numbers (relatively prime).

Numbers have no common prime factors.

### 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".