1 December 2023

The LCM is a widely used concept in the field of mathematics, and we may need it to solve anything from a simple sum of fractions to algebraic expressions. In itself, it consists of get the smallest multiple of a number.

If you want to know more about this interesting topic, keep reading! Here we will explain from what the proper and prime factors are to how to obtain the LCM and what it is for. This in order to be able to give you a wide knowledge in the world of exact sciences.

## Concept and definition of the least common multiple in fractions

The least common multiple (LCM) is a mathematical and trigonometric function that refers to the smallest number, and non-zero, which is a multiple of a set of natural numbers. To better understand this concept, it is important to define some terms:

• a multiple It is any number obtained by multiplying one number by another, and can be positive or negative. For example, the multiples of the number 2 are 2, 4, 6, 8, etc.
• a common multiple is a number that is a multiple of two or more digits at the same time. For example, if we want to find the common multiples of 2 and 3, we must calculate the multiples of each number (2, 4, 6, 8, 10, etc. for 2; and 3, 6, 9, 12, etc. for 3 ) and find the ones they have in common (6, 12, etc.).

The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, and so on; and, the multiples of 3, would be 3, 6, 9, 12, 15, 18, 21, and so on. As you can see, the common multiples of 2 and 3 they are the numbers 6, 12 and 18, and there are infinitely many common multiples.

The least common multiple (LCM) is the smallest number that is a common multiple of a set of figures, and can be obtained by finding the common multiples and selecting the smallest. For example, in the case of the numbers 2 and 3, the common multiples are 6, 12, and 18; and, as such, the LCM is 6, since it is the smallest number among the common multiples.

The LCM is useful for adding fractions with unlike denominators, and to do this, one must find the LCM of the denominators of the fractions and convert them into equivalent fractions with the same denominator. Then the fractions can be easily added.

## What is the least common multiple used for?

The LCM is a highly useful mathematical tool for finding the smallest number that is common multiple of several natural numbers. This is used, for example, for the addition of fractions with different denominators, where the LCM of the denominators is sought to be able to convert the fractions to equivalent fractions with the same denominator and, in such a way, to be able to add them.

Also used the LCM in algebraic expressions, since the LCM of two algebraic expressions refers to the one with a smaller numerical coefficient and a lower degree. This makes it susceptible to dividing by the given expressions without leaving any out.

## How do you get the least common multiple?

Here are the steps so you know how take the least common multiple:

• First, to get the LCM of two numbers we need to decompose them into ‘prime components’. Therefore, the LCM will be the figure we get from the multiplication of common factors and not generals, choosing the one with the elevation to the highest power.
• To carry out this operation it is recommended to draw a vertical line as follows: to the left of the row we will place the amount of which we are going to get the LCM, and to the right of the line we must put the prime data. This will be the one that will divide the figure from which we will obtain the LCM.
• The first thing to do is break down each of the digits into prime aspects. Then we will choose the common elements and non-generals raised to the highest exponent. Finally, we must multiply the selected data.

In this way, we achieve get the least common multiple.

## What is the relationship between the least common multiple and the greatest common divisor?

Both are values ​​obtained by counting the divided factors of two or more numbers. That is, the LCM and the GCD are calculated from the same information, but they are interpreted differently.

We can highlight that one of their similarities is that the data that is multiplied and the divisors are related to each other. Like almost everything in the field of mathematics, most of them are consistent, since in both concepts we are using usual numbers and not general ones.

## Examples of how the least common multiple is calculated

Next is will show some examples how to calculate the LCM:

• Let’s calculate the LCM of 4 and 6

First we break down in prime factors:

• 4 2 6 2
• 2 2 3 3
• eleven

then we choose the usual and not general data, taking the one with the largest exponent. Then multiply:

• 4=2 2
• 6= 3×2
• 4=2x2x3x2=24

The least common multiple of 4 and 6 is 24.

• Now we are going to calculate the LCM of 12 and 8:

we decompose in prime factors:

• 12 2 8 2
• 6 2 4 2
• 3 3 2 2
• eleven

we choose common factors and multiply:

12=22 x3

8=23

2x2x2x3= 24

So the LCM of 12 and 8 is 24.

we decompose in prime factors:

• 40 2 16 2
• 20 2 8 2
• 10 2 4 2
• 5 5 2 2
• eleven

We separate the common factors and multiply them:

• 40=23×5
• 16= 24
• 2x2x2x2x5=80

We have the result that LCM of 40 and 16 is 80.

We know that the least common multiple can be used when solving addition in fractions. With lots of practice and review constant on this subject, you will become an expert! After all, you should keep in mind that calculations, believe it or not, are present in our daily lives.