Least common multiple

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Least common multiple

Least common multiple

The least common multiple of two or more non-zero whole numbers is the smallest number that can divide those numbers.

To find the least common multiple, we will use a method in which we list all the multiples of the numbers in question, and then find the lowest common multiple for those numbers. The lowest common multiple of numbers $a$ and $b$ is denoted as LCM (a, b).

We will now move on to an example. We need to find the least common multiple of numbers $9$ and $30$. We will get the multiples of the numbers by multiplying them with numbers $2, 3, 4, 5, 6, …$.

Multiples of number $30$ are: $60, 90, 120, 150, 180…$.
Multiples of number $9$ are: $18, 27, 36, 45, 54, 63, 72, 81, 90, 99…$.
The lowest number that can be created by multiplying the numbers $30$ and $9$ with other numbers and that is a multiple of them both is the number $90$. That means that the lowest common multiple of the numbers $30$ and $9$ is the number $90$.

lcm of numbers 9 and 30

The method described above can be used to find the lowest common multiple of three numbers as well. For example, now we will find the LCM of numbers $6$, $9$ and $12$.

Multiples of number $6$ are numbers: $12, 18, 24, 30, 36, 42…$
Multiples of number $9$ are numbers: $18, 27, 36, 45…$
Multiples of number $12$ are numbers: $24, 36, 48, 60…$
The lowest multiple that all three numbers have in common is number $36$. So, number $36$ is the LCM of numbers $6$, $9$ and $12$.

lcm of numbers 6, 9 and 12

Another methods

There is also another method for finding the lowest common multiple. In this method we need to divide the numbers into prime factors through a process called factorization.

Then we need to find the largest count for each of the prime factors of those numbers and write them down. After that, we just need to multiply those numbers by themselves the largest number of times they appear in any of the factorizations, then multiply all those values together and we will get the lowest common multiple. Seems complicated? It really isn’t. Just take a look at the following examples.

We’ll find the lowest common multiple of numbers from our previous examples using this new method. For example, we will find the lowest common multiple of numbers $9$ and $30$ by going through the following steps:

Example:

Find the prime factors of numbers $9$ and $30$.

prime factors of number 9

The prime factors of the number $9$ are because $9 = 3 \cdot 3$.

prime factors of number 30

The prime factors of the number $30$ are numbers $2$, $3$ and $5$.

Now we need to count the number of times each factor of the numbers appears in the factorization.

For the number $9$, the number $3$ appears twice. For number $30$ the factors $2$, $3$ and $5$ appears once.

In this step we need to write all the different factors of numbers $30$ and $9$ and write down the biggest number of appearances of that factor. The number $2$ “appears once” in number $30$. The number $3$ appears once in number $30$ and twice in number $9$. We need to write the biggest number of appearances down, and that is twice. The number $5$ appears once in number $30$.

Now we only need to multiply the numbers we have written down in the previous step. The lowest common multiple is the product of those numbers.

lcm equals 90

The number $3$ appears twice, so we had to multiply it by itself twice (as in $3\cdot3$) in the final result of the lowest common multiple. Numbers $2$ and $5$ appeared only once and we only had to multiply them once.

Example:

Now we are going to find the lowest common multiple of numbers $6$, $9$ and $12$.

Solution:

Step 1: find the prime factors.

prime factors of three numbers

Step 2: write down the biggest number of appearances of factors in numbers $6$, $9$ and $12$. The number $2$ appears twice. The number $3$ appears twice as well.

Step 3: find the lowest common multiple by multiplying the numbers from the previous step.

lcm equals 36

Both of the methods are fairly simple and easy to use. You will need to brush up a bit on your factorization skills, but the rest is just following the steps described above.