# Are the Two Numbers 34 and 9 Coprime (Relatively Prime, Prime to Each Other)? Check if Their Greatest Common Factor, GCF, Is Equal to 1

## Are the numbers 34 and 9 coprime (prime to each other, relatively prime)?

### 34 and 9 are coprime (relatively prime)... if:

#### If there is no number other than 1 that evenly divides (without a remainder) both numbers. Or...

#### Or, in other words, if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is 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.

#### 34 = 2 × 17

34 is not a prime number, is a composite one.

#### 9 = 3^{2}

9 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).

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

### gcf (hcf, gcd) (34; 9) = 1

Coprime numbers (prime to each other, relatively prime)

## Coprime numbers (prime to each other, relatively prime) (34; 9)? Yes.

The numbers have no common prime factors.

gcf (hcf, gcd) (9; 34) = 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:

34 ÷ 9 = 3 + 7

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

9 ÷ 7 = 1 + 2

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

7 ÷ 2 = 3 + 1

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

2 ÷ 1 = 2 + 0

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

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

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

### gcf (hcf, gcd) (34; 9) = 1

## Coprime numbers (prime to each other, relatively prime) (34; 9)? Yes.

gcf (hcf, gcd) (9; 34) = 1

## Other similar operations with coprime numbers:

## Are the two numbers coprime (relatively prime)?

#### Two natural numbers are coprime (relatively prime, prime to each other) - if there is no number that is evenly dividing both numbers (= without a remainder), that is, if their greatest (highest) common factor (divisor), gcf, or hcf, or gcd is 1.

#### Two natural numbers are not relatively prime - if there is at least one number that evenly divides the two numbers, that is, if their greatest common factor, gcf, is not 1.