GCF of 72 and 84


The gcf of 72 and 84 is the largest positive integer that divides the numbers 72 and 84 without a remainder. Spelled out, it is the greatest common factor of 72 and 84. Here you can find the gcf of 72 and 84, along with a total of three methods for computing it. In addition, we have a calculator you should check out. Not only can it determine the gcf of 72 and 84, but also that of three or more integers including seventy-two and eighty-four for example. Keep reading to learn everything about the gcf (72,84) and the terms related to it.

What is the GCF of 72 and 84

If you just want to know what is the greatest common factor of 72 and 84, it is 12. Usually, this is written as

gcf(72,84) = 12

The gcf of 72 and 84 can be obtained like this:

  • The factors of 72 are 72, 36, 24, 18, 12, 9, 8, 6, 4, 3, 2, 1.
  • The factors of 84 are 84, 42, 28, 21, 14, 12, 7, 6, 4, 3, 2, 1.
  • The common factors of 72 and 84 are 12, 6, 4, 3, 2, 1, intersecting the two sets above.
  • In the intersection factors of 72 ∩ factors of 84 the greatest element is 12.
  • Therefore, the greatest common factor of 72 and 84 is 12.

Taking the above into account you also know how to find all the common factors of 72 and 84, not just the greatest. In the next section we show you how to calculate the gcf of seventy-two and eighty-four by means of two more methods.

How to find the GCF of 72 and 84

The greatest common factor of 72 and 84 can be computed by using the least common multiple aka lcm of 72 and 84. This is the easiest approach:

gcf (72,84) = $\frac{72 \times 84}{lcm(72,84)} = \frac{6048}{504}$ = 12

Alternatively, the gcf of 72 and 84 can be found using the prime factorization of 72 and 84:

  • The prime factorization of 72 is: 2 x 2 x 2 x 3 x 3
  • The prime factorization of 84 is: 2 x 2 x 3 x 7
  • The prime factors and multiplicities 72 and 84 have in common are: 2 x 2 x 3
  • 2 x 2 x 3 is the gcf of 72 and 84
  • gcf(72,84) = 12

In any case, the easiest way to compute the gcf of two numbers like 72 and 84 is by using our calculator below. Note that it can also compute the gcf of more than two numbers, separated by a comma. For example, enter 72,84. The calculation is conducted automatically.

The gcf is...
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Use of GCF of 72 and 84

What is the greatest common factor of 72 and 84 used for? Answer: It is helpful for reducing fractions like 72 / 84. Just divide the nominator as well as the denominator by the gcf (72,84) to reduce the fraction to lowest terms.

$\frac{72}{84} = \frac{\frac{72}{12}}{\frac{84}{12}} = \frac{6}{7}$.

Properties of GCF of 72 and 84

The most important properties of the gcf(72,84) are:

  • Commutative property: gcf(72,84) = gcf(84,72)
  • Associative property: gcf(72,84,n) = gcf(gcf(84,72),n) $\hspace{10px}n\hspace{3px}\epsilon\hspace{3px}\mathbb{Z}$

The associativity is particularly useful to get the gcf of three or more numbers; our calculator makes use of it.

To sum up, the gcf of 72 and 84 is 12. In common notation: gcf (72,84) = 12.

If you have been searching for gcf 72 and 84 or gcf 72 84 then you have come to the correct page, too. The same is the true if you typed gcf for 72 and 84 in your favorite search engine.

Note that you can find the greatest common factor of many integer pairs including seventy-two / eighty-four by using the search form in the sidebar of this page.

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