LCM of 3 and 9


The lcm of 3 and 9 is the smallest positive integer that divides the numbers 3 and 9 without a remainder. Spelled out, it is the least common multiple of 3 and 9. Here you can find the lcm of 3 and 9, 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 9, but also that of three or more integers including three and nine for example. Keep reading to learn everything about the lcm (3,9) and the terms related to it.

What is the LCM of 3 and 9

If you just want to know what is the least common multiple of 3 and 9, it is 9. Usually, this is written as

lcm(3,9) = 9

The lcm of 3 and 9 can be obtained like this:

  • The multiples of 3 are …, 6, 9, 12, ….
  • The multiples of 9 are …, 0, 9, 18, …
  • The common multiples of 3 and 9 are n x 9, intersecting the two sets above, $\hspace{3px}n \hspace{3px}\epsilon\hspace{3px}\mathbb{Z}$.
  • In the intersection multiples of 3 ∩ multiples of 9 the least positive element is 9.
  • Therefore, the least common multiple of 3 and 9 is 9.

Taking the above into account you also know how to find all the common multiples of 3 and 9, not just the smallest. In the next section we show you how to calculate the lcm of three and nine by means of two more methods.

How to find the LCM of 3 and 9

The least common multiple of 3 and 9 can be computed by using the greatest common factor aka gcf of 3 and 9. This is the easiest approach:

lcm (3,9) = $\frac{3 \times 9}{gcf(3,9)} = \frac{27}{3}$ = 9

Alternatively, the lcm of 3 and 9 can be found using the prime factorization of 3 and 9:

  • The prime factorization of 3 is: 3
  • The prime factorization of 9 is: 3 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) = 9

In any case, the easiest way to compute the lcm of two numbers like 3 and 9 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 3,9. Push the button only to start over.

The lcm is...
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Use of LCM of 3 and 9

What is the least common multiple of 3 and 9 used for? Answer: It is helpful for adding and subtracting fractions like 1/3 and 1/9. Just multiply the dividends and divisors by 3 and 1, respectively, such that the divisors have the value of 9, the lcm of 3 and 9.

$\frac{1}{3} + \frac{1}{9} = \frac{3}{9} + \frac{1}{9} = \frac{4}{9}$. $\hspace{30px}\frac{1}{3} – \frac{1}{9} = \frac{3}{9} – \frac{1}{9} = \frac{2}{9}$.

Properties of LCM of 3 and 9

The most important properties of the lcm(3,9) are:

  • Commutative property: lcm(3,9) = lcm(9,3)
  • Associative property: lcm(3,9,n) = lcm(lcm(9,3),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 9 is 9. In common notation: lcm (3,9) = 9.

If you have been searching for lcm 3 and 9 or lcm 3 9 then you have come to the correct page, too. The same is the true if you typed lcm for 3 and 9 in your favorite search engine.

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

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