The GCF (1,800 and 120) = ? Calculate the Greatest (Highest) Common Factor (Divisor), GCF (HCF, GCD), of the Numbers, by Two Methods: 1) The Numbers' Divisibility and 2) The Prime Factorization

gcf, hcf, gcd (1,800; 120) = ?

Method 1. The divisibility of numbers:

Divide the larger number by the smaller one.

Note that when the numbers are divided, the remainder is zero:

1,800 ÷ 120 = 15 + 0

⇒ 1,800 = 120 × 15

So, 1,800 is divisible by 120.

And 120 is a factor (divisor) of 1,800.

Also, the greatest factor (divisor) of 120 is the number itself, 120.

» Online calculator. The divisibility of numbers


The greatest (highest) common factor (divisor),
gcf, hcf, gcd (1,800; 120) = 120 = 23 × 3 × 5
1,800 is divisible by 120
Scroll down for the 2nd method...

Method 2. The prime factorization:

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

1,800 = 23 × 32 × 52
1,800 is not a prime number but a composite one.

120 = 23 × 3 × 5
120 is not a prime number but a composite one.

» Online calculator. Check whether a number is prime or not. The prime factorization of composite numbers

* Prime number: a natural number that is only divisible by 1 and itself. A prime number has exactly 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):

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


The greatest (highest) common factor (divisor),
gcf, hcf, gcd (1,800; 120) = 23 × 3 × 5 = 120
1,800 contains all the prime factors of the number 120
1,800 is divisible by 120.

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 similar operations with the greatest (highest) common factor (divisor):

» What is the greatest (highest) common factor (divisor) (gcf, gcd, hcf) of the numbers 1,787 and 116?

» What is the greatest (highest) common factor (divisor) (gcf, gcd, hcf) of the numbers 1,813 and 121?


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.

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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.

  • Note 1: The greatest common factor (gcf) is also called the highest common factor (hcf), or the greatest common divisor (gcd).
  • Note 2: The Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
  • Suppose the number "t" evenly divides the number "a" ( = when evenly dividing the number "a" by "t", the remainder is zero).
  • When we look at the prime factorization of "a" and "t", we find that:
  • 1) all the prime factors of "t" are also prime factors of "a"
  • and
  • 2) the exponents of the prime factors of "t" are equal to or smaller than the exponents of the prime factors of "a" (see the * Note below)
  • For example, the number 12 is a divisor (a factor) of the number 60:
  • 12 = 2 × 2 × 3 = 22 × 3
  • 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
  • * Note: 23 = 2 × 2 × 2 = 8. We say that 2 was raised to the power of 3. In this example, 3 is the exponent and 2 is the base. The exponent indicates how many times the base is multiplied by itself. 23 is the power and 8 is the value of the power.
  • If the number "t" is a common divisor of the numbers "a" and "b", then:
  • 1) "t" only has the prime factors that also intervene in the prime factorization of "a" and "b".
  • and
  • 2) each prime factor of "t" has the smallest exponents with respect to the prime factors of the numbers "a" and "b".
  • For example, the number 12 is the common divisor of the numbers 48 and 360. Below is their prime factorization:
  • 12 = 22 × 3
  • 48 = 24 × 3
  • 360 = 23 × 32 × 5
  • You can see that the number 12 has only the prime factors that also occur in the prime factorization of the numbers 48 and 360.
  • You can see above that the numbers 48 and 360 have several common factors: 2, 3, 4, 6, 8, 12, 24. Out of these, 24 is the greatest common factor (GCF) of 48 and 360.
  • 24 = 2 × 2 × 2 × 3 = 23 × 3
  • 48 = 24 × 3
  • 360 = 23 × 32 × 5
  • 24, the greatest common factor of the numbers 48 and 360, is calculated as the product of all the common prime factors of the two numbers, taken by the smallest exponents (powers).
  • If two numbers "a" and "b" have no other common factor than 1, gcf (a, b) = 1, then the numbers "a" and "b" are called coprime numbers (relatively prime, prime to each other).
  • If "a" and "b" are not relatively prime numbers, then every common divisor of "a" and "b" is a divisor of the greatest common divisor of "a" and "b".
  • Let's have an example on how to calculate the greatest common factor, gcf, of the following numbers:
  • 1,260 = 22 × 32
  • 3,024 = 24 × 32 × 7
  • 5,544 = 23 × 32 × 7 × 11
  • gcf (1,260, 3,024, 5,544) = 22 × 32 = 252
  • And another example:
  • 900 = 22 × 32 × 52
  • 270 = 2 × 33 × 5
  • 210 = 2 × 3 × 5 × 7
  • gcf (900, 270, 210) = 2 × 3 × 5 = 30
  • And one more example:
  • 90 = 2 × 32 × 5
  • 27 = 33
  • 22 = 2 × 11
  • gcf (90, 27, 22) = 1 - The three numbers have no prime factors in common, they are relatively prime.