### 5,382 and 1,998 are not relatively prime... if:

#### If there is at least one number other than 1 that evenly divides the two numbers (without a remainder). Or...

#### Or, in other words, if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is not equal to 1.

## Calculate the greatest (highest) common factor (divisor),

gcf (hcf, gcd), of the two numbers

### Method 1. The prime factorization:

#### The prime factorization of a number: finding the prime numbers that multiply together to make that number.

#### 5,382 = 2 × 3^{2} × 13 × 23

5,382 is not a prime number, is a composite one.

#### 1,998 = 2 × 3^{3} × 37

1,998 is not a prime number, is a composite one.

#### Prime number: a number that is divisible (dividing evenly) only by 1 and itself. A prime number has only two factors: 1 and itself.

#### Composite number: a natural number that has at least one other factor than 1 and itself.

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

#### Multiply all the common prime factors of the two numbers, taken by their smallest exponents (powers).

### gcf (hcf, gcd) (5,382; 1,998) = 2 × 3^{2} = 18 ≠ 1

## Coprime numbers (prime to each other, relatively prime) (5,382; 1,998)? No.

The two numbers have common prime factors.

gcf (hcf, gcd) (1,998; 5,382) = 18 ≠ 1

Scroll down for the 2nd method...

### Method 2. The Euclidean Algorithm:

#### This algorithm involves the process of dividing numbers and calculating the remainders.

#### 'a' and 'b' are the two natural numbers, 'a' >= 'b'.

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

#### If 'r' = 0, STOP. 'b' = the gcf (hcf, gcd) of 'a' and 'b'.

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

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

5,382 ÷ 1,998 = 2 + 1,386

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

1,998 ÷ 1,386 = 1 + 612

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

1,386 ÷ 612 = 2 + 162

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

612 ÷ 162 = 3 + 126

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

162 ÷ 126 = 1 + 36

Step 6. Divide the remainder of the step 4 by the remainder of the step 5:

126 ÷ 36 = 3 + 18

Step 7. Divide the remainder of the step 5 by the remainder of the step 6:

36 ÷ 18 = 2 + 0

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

18 is the number we were looking for - the last non-zero remainder.

This is the greatest (highest) common factor (divisor).

### gcf (hcf, gcd) (5,382; 1,998) = 18 ≠ 1

## Coprime numbers (prime to each other, relatively prime) (5,382; 1,998)? No.

gcf (hcf, gcd) (1,998; 5,382) = 18 ≠ 1