gcf (6,551; 135) = ? Calculate the greatest (highest) common factor (divisor) of numbers, gcf (hcf, gcd), by two methods: 1) The prime factorization and 2) The Euclidean Algorithm

gcf, hcf, gcd (6,551; 135) = ?

Method 1. The prime factorization:

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


6,551 is a prime number and cannot be broken down into other prime factors.


135 = 33 × 5
135 is not a prime number but a composite one.


* The natural numbers that are only divisible by 1 and themselves are called prime numbers. A prime number has exactly two factors: 1 and itself.
* A composite number is a natural number that has at least one other factor than 1 and itself.



Calculate the greatest (highest) common factor (divisor):

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


But the two numbers have no common prime factors.


gcf, hcf, gcd (6,551; 135) = 1



gcf, hcf, gcd (6,551; 135) = 1
Coprime numbers (prime to each other, relatively prime).

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:
6,551 ÷ 135 = 48 + 71
Step 2. Divide the smaller number by the above operation's remainder:
135 ÷ 71 = 1 + 64
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
71 ÷ 64 = 1 + 7
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
64 ÷ 7 = 9 + 1
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
7 ÷ 1 = 7 + 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).


The greatest (highest) common factor (divisor):
gcf, hcf, gcd (6,551; 135) = 1


gcf, hcf, gcd (6,551; 135) = 1
Coprime numbers (prime to each other, relatively prime).

The final answer:
The greatest (highest) common factor (divisor),
gcf, hcf, gcd (6,551; 135) = 1
Coprime numbers (prime to each other, relatively prime).
The two numbers have no prime factors in common.

Why do we need to calculate the greatest common factor?

Once you've calculated the greatest common factor of the numerator and the denominator of a fraction, it becomes much easier to fully reduce (simplify) the fraction to the lowest terms (the smallest possible numerator and denominator).



Other operations of the same kind:


Calculator of the greatest (highest) common factor (divisor), gcf, hcf, gcd

Calculate the greatest (highest) common factor (divisor) of numbers, gcd, hcf, gcd:

Method 1: Run the prime factorization of the numbers - then multiply all the common prime factors, taken by their smallest exponents. If there are no common prime factors, then gcf equals 1.

Method 2: The Euclidean Algorithm.

Method 3: The divisibility of the numbers.

The greatest (highest) common factor (divisor), gcf (hcf, gcd): the latest calculated

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The greatest (highest) common factor (divisor), gcf (hcf, gcd): the list of all the calculations

The greatest (highest) common factor (divisor), gcf, hcf, gcd. What it is and how to calculate it.


What is a prime number? Definition, examples

What is a composite number? Definition, examples

The prime numbers up to 1,000

The prime numbers up to 10,000

The Sieve of Eratosthenes

The Euclidean Algorithm

Completely reduce (simplify) fractions to the lowest terms: Steps and Examples