# Are the Two Numbers 1,278 and 4,236 Coprime (Relatively Prime, Prime to Each Other)? Check if Their Greatest Common Factor, GCF, Is Equal to 1

## Are the numbers 1,278 and 4,236 coprime (prime to each other, relatively prime)?

### 1,278 and 4,236 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.

#### 1,278 = 2 × 3^{2} × 71

1,278 is not a prime number, is a composite one.

#### 4,236 = 2^{2} × 3 × 353

4,236 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) (1,278; 4,236) = 2 × 3 = 6 ≠ 1

## Coprime numbers (prime to each other, relatively prime) (1,278; 4,236)? No.

The two numbers have common prime factors.

gcf (hcf, gcd) (1,278; 4,236) = 6 ≠ 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:

4,236 ÷ 1,278 = 3 + 402

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

1,278 ÷ 402 = 3 + 72

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

402 ÷ 72 = 5 + 42

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

72 ÷ 42 = 1 + 30

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

42 ÷ 30 = 1 + 12

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

30 ÷ 12 = 2 + 6

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

12 ÷ 6 = 2 + 0

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

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

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

### gcf (hcf, gcd) (1,278; 4,236) = 6 ≠ 1

## Coprime numbers (prime to each other, relatively prime) (1,278; 4,236)? No.

gcf (hcf, gcd) (1,278; 4,236) = 6 ≠ 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.