### 1,377 and 1,189 are coprime (relatively, mutually prime) if they have no common prime factors, that is, if their greatest (highest) common factor (divisor), gcf, hcf, gcd, is 1.

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

### Approach 1. Integer numbers prime factorization:

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

#### 1,377 = 3^{4} × 17;

1,377 is not a prime, is a composite number;

#### 1,189 = 29 × 41;

1,189 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 greatest (highest) common factor (divisor):

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

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

#### gcf, hcf, gcd (1,377; 1,189) = 1;

coprime numbers (relatively prime)

## Coprime numbers (relatively prime) (1,377; 1,189)? Yes.

Numbers have no common prime factors.

gcf, hcf, gcd (1,189; 1,377) = 1.

### 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:

1,377 ÷ 1,189 = 1 + 188;

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

1,189 ÷ 188 = 6 + 61;

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

188 ÷ 61 = 3 + 5;

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

61 ÷ 5 = 12 + 1;

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

5 ÷ 1 = 5 + 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).

#### gcf, hcf, gcd (1,377; 1,189) = 1;

## Coprime numbers (relatively prime) (1,377; 1,189)? Yes.

gcf, hcf, gcd (1,189; 1,377) = 1.