Maximum Common Divider – mcd

Let’s explain what the maximum common divisor method is for calculating it and when it is used, with exercises solved step by step to make it completely clear to you.

Divisors of a number

The divisors of a number are those that when dividing that number, the result is exact. For example:

9:3 = 3

We say that 3 is a divisor of 9 and in turn, 9 is a multiple of 3.

In order to obtain the divisors of a number, the Divisibility Rules are applied, which you can consult if you click on the link.

What is the Maximum Common Divisor (m.c.d.)

The maximum common divisor of 2 or more numbers, is a divisor that is common to those numbers and is also the highest of the divisors that are common.

For example, let’s look at it with the divisors of 20 and 10:

Divisors of 20: 2, 4, 5, 10, 20

Dividers of 10: 2, 5, 10

The divisors highlighted in blue are the common divisors that have 10 and 20.

Unlike multiples, which are infinite, the divisors are a concrete quantity and in this case, the common divisors are 2, 5 and 10.

The highest of the three is 10, so 10 is the maximum common divisor of 10 and 20.

How to Calculate the Maximum Common Divider (m.c.d.)

The procedure for calculating it is as follows:

1 – The numbers are broken down into prime factors.

2- Only the common factors that appear when the numbers are broken down, elevated to the lowest exponent, are selected.

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

Resolved exercises to calculate Maximum Common Divider step by step

Resolved exercise 1

Calculate the maximum common divisor of 30 and 36

1 – The numbers are broken down into prime factors:


2 – We choose only the common factors. In 30 we have a 2, a 3 and a 5. In 36 we have a 2 and a 3. Let’s put the two decompositions together to see it clearer:


Therefore the chosen factors are 2 and 3. 5 is not common because 36 does not have it.

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

The 2 we have raised to 1 and 2, so we keep the 1

The 3 we have raised to 1 and 2, so we keep the 1

The factors elevated to their lowest exponents would be 2 and 3.

The M.C.D. of 30 and 36 is the multiplication of these factors with their exponents:


Notice that we only chose the factors that were common to the two numbers. We didn’t choose 5. If it is not common, it is not chosen.

Once the factors have been chosen, the lowest exponent is selected for each one.

Is the method clear to you? We have to choose the common factors and then for each factor we have to choose the lowest exponent.

Resolved exercise 2

Calculate the maximum common divisor of 15, 50 and 125

1 – We decompose the numbers:


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


Only the 5 is common to 15, 50 and 125. Then we choose the 5.

The least exponent of 5 is 1

The factor chosen with its lowest exponent would be 5.

The M.C.D. of 15, 50 and 125 would be in this case only this factor:


Be careful, we may not have several numbers and there is no factor in common to them. In that case there will be no common maximum divisor.

For example: Calculate the maximum common divisor of 6, 10 and 15.

1 – We decompose the numbers:


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


There is no factor that is common to all three numbers. So we can’t choose any of them.

Since we do not have a factor, we cannot choose its exponent either.

The M.C.D. of 6, 10 and 15 would not exist:


The minimum common multiple of two or more numbers, by contrast, always exists, because as a minimum it will be the multiplication between them.

Resolved exercise 3

Calculate the maximum common divisor of 6, 12 and 30.

We break it down into prime factors:

how to calculate the maximum common divisor

We choose the factors common to the 3 numbers. We see that the 2 and the 3 are repeated in the 3 and the smallest exponent to which each of them is found is the 1. Therefore we choose 2 and 3.

The maximum common divisor will be:

how to get the maximum common divisor

If there was no common factor, there would be no maximum common divisor of those numbers.