LCM of 3 and 12


The lcm of 3 and 12 is the smallest positive integer that divides the numbers 3 and 12 without a remainder. Spelled out, it is the least common multiple of 3 and 12. Here you can find the lcm of 3 and 12, 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 lcm of 3 and 12, but also that of three or more integers including three and twelve for example. Keep reading to learn everything about the lcm (3,12) and the terms related to it.

What is the LCM of 3 and 12

If you just want to know what is the least common multiple of 3 and 12, it is 12. Usually, this is written as

lcm(3,12) = 12

The lcm of 3 and 12 can be obtained like this:

  • The multiples of 3 are …, 9, 12, 15, ….
  • The multiples of 12 are …, 0, 12, 24, …
  • The common multiples of 3 and 12 are n x 12, intersecting the two sets above, $\hspace{3px}n \hspace{3px}\epsilon\hspace{3px}\mathbb{Z}$.
  • In the intersection multiples of 3 ∩ multiples of 12 the least positive element is 12.
  • Therefore, the least common multiple of 3 and 12 is 12.

Taking the above into account you also know how to find all the common multiples of 3 and 12, not just the smallest. In the next section we show you how to calculate the lcm of three and twelve by means of two more methods.

How to find the LCM of 3 and 12

The least common multiple of 3 and 12 can be computed by using the greatest common factor aka gcf of 3 and 12. This is the easiest approach:

lcm (3,12) = $\frac{3 \times 12}{gcf(3,12)} = \frac{36}{3}$ = 12

Alternatively, the lcm of 3 and 12 can be found using the prime factorization of 3 and 12:

  • The prime factorization of 3 is: 3
  • The prime factorization of 12 is: 2 x 2 x 3
  • Eliminate the duplicate factors of the two lists, then multiply them once with the remaining factors of the lists to get lcm(3,3) = 12

In any case, the easiest way to compute the lcm of two numbers like 3 and 12 is by using our calculator below. Note that it can also compute the lcm of more than two numbers, separated by a comma. For example, enter 3,12. Push the button only to start over.

The lcm is...
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Use of LCM of 3 and 12

What is the least common multiple of 3 and 12 used for? Answer: It is helpful for adding and subtracting fractions like 1/3 and 1/12. Just multiply the dividends and divisors by 4 and 1, respectively, such that the divisors have the value of 12, the lcm of 3 and 12.

$\frac{1}{3} + \frac{1}{12} = \frac{4}{12} + \frac{1}{12} = \frac{5}{12}$. $\hspace{30px}\frac{1}{3} – \frac{1}{12} = \frac{4}{12} – \frac{1}{12} = \frac{3}{12}$.

Properties of LCM of 3 and 12

The most important properties of the lcm(3,12) are:

  • Commutative property: lcm(3,12) = lcm(12,3)
  • Associative property: lcm(3,12,n) = lcm(lcm(12,3),n) $\hspace{10px}n\neq 0 \hspace{3px}\epsilon\hspace{3px}\mathbb{Z}$

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

To sum up, the lcm of 3 and 12 is 12. In common notation: lcm (3,12) = 12.

If you have been searching for lcm 3 and 12 or lcm 3 12 then you have come to the correct page, too. The same is the true if you typed lcm for 3 and 12 in your favorite search engine.

Note that you can find the least common multiple of many integer pairs including three / twelve by using the search form in the sidebar of this page.

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