We are going to explain how the **minimum common multiple** is obtained and when it is used. We’ll also solve a step-by-step example so that it’s completely clear to you.

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## Multiples of a number

The result of multiplying that number by any other natural number is called a multiple of a number. Since natural numbers are infinite, multiples of a number are also infinite.

They are multiples of 5: 5, 10,15, 20, 25, 30… since they result from multiplying 5 by the rest of natural numbers:

5 x 1 = 5

5 x 2 = 10

5 x 3 = 15

5 x 4 = 20

Thus, multiples of any number can be obtained:

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27 ….

Multiples of 9: 9, 18, 27, 36, 45, 54, 63 ….

Remember: Multiples of a number are always greater than that number

## What is the minimum common multiple (mcm)

The minimum common multiple of 2 or more numbers is the multiple that is common to those numbers and is also the lowest. For example, let’s see it with 3 and 9:

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27 ….

Multiples of 9: 9, 18, 27, 36, 45, 54, 63 ….

We see that they have in common as multiples 9, 18, 27… The smallest of all of them is 9, therefore, that will be the minimum common multiple of 3 and 9.

Used to calculate the lowest common denominator when adding or subtracting fractions.

## How to calculate the minimum common multiple

The method for calculating the lowest common multiple most directly is as follows:

1 – Numbers are broken down into prime factors.

2- All the factors that appear when decomposing the numbers, common and uncommon, elevated to the highest exponent are selected

3 – The chosen factors are multiplied (with the chosen exponent for each one) and the result will be the minimum common multiple

## Calculation example of Common Minimum Multiple step by step

### Resolved exercise 1

We will use the same example as before. We are going to calculate the minimum common multiple of 3 and 9, which is written like this:

m.c.m. (3 and 9)

1 – First we break down the numbers. The 3 is not necessary, since it is primo.

Thus we break down 9:

2 – Then we choose the factors. We write 3 and 9 but decomposed in prime factors:

3 = 3

9 = 3²

The only factor is 3.

The maximum exponent to which it is elevated is 2.

Therefore, it remains as 3².

Finally, we perform the power:

Let’s see another somewhat more complicated example. Find the minimum common multiple of 6, 12 and 30.

First, we break down into prime factors:

Now we choose the factors:

We have 2, 3 and 5.

The largest exponent for 2 is a 2. 3 and 5 are elevated to 1.

We therefore chose 2² 3 and 5.

In the last step, we redo the powers and multiply:

### Resolved exercise 2

Calculate the minimum common multiple of 4, 24, and 36

1 – Numbers are broken down into prime factors:

2 – We choose the common and uncommon factors. In 4 we have a 2, in 24 we have a 2 and a 3 and in 36 another 2 and another 3. Let’s put the 3 decompositions together to see it clearer:

Thus the factors chosen are 2 and 3.

Now we are going to choose the exponents, which will be the largest for each one:

We have elevated 2 to 2 and 3, so we keep 3

We have raised 3 to 1 and 2, so we keep 2

The factors elevated to their greatest exponents would be 2³ and 3².

The m.c.m. of 4, 24 and 36 is the multiplication of these factors with their exponents:

Is the method clear to you? Only if a factor appears once in the decompositions, you have to choose it and then for each factor you have to choose the highest exponent

### Resolved exercise 3

Calculate the minimum common multiple of 12, 45, and 50

1 – We break down the numbers:

2 – We choose the common and uncommon factors. These are the decomposed numbers:

We have 2, 3 and 5 (each of them appears at least once)

The greatest exponent of 2 is 2

The highest exponent of 3 is 2

The highest exponent of 5 is 2

The factors elevated to their greatest exponents would be 2², 3² and 5².

The m.c.m. of 12, 45 and 50 is the multiplication of these factors with their exponents:

You’ll wonder: And what applications does the least common multiple have? When am I going to use it?

Then m.c.m. is mostly used to calculate the common denominator, when you have to add or subtract fractions with different denominator.