Minimum common multiple – m.c.m.

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.

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:

how to obtain the minimum common multiplo

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:

what the mcm is and how it is obtained

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:

how the minimum common multiplo is calculated

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:

which is and how the minimum common multiple is obtained

Resolved exercise 2

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

1 – Numbers are broken down into prime factors:

mcm-1

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:

mcm-2

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:

mcm-3

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:

mcm-4

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

mcm-5

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:

mcm-6

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.

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