Greatest Common Divisor and Lowest Common Multiple between Monomials

The Greatest Common Divisor of Monomials is that monomial of maximum degree which simultaneously divides all given monomials.

Similarly, the Lowest Common Multiple of Monomials is that monomial of lowest degree which is simultaneously divisible by all given monomials.

In this lesson, we will look at how to calculate the GCD between monomials and the lcm between monomials from the definition of divisibility between monomials. We will consider how to calculate the GCD and the lcm, starting with the case of two monomials, in order to obtain the general rule for calculating GCD of several monomials and lcm of several monomials.

Greatest Common Divisor between monomials

In general, we can define the Maximum Common Divisor between Monomials as any monomial of maximum degree that simultaneously divides all given monomials.

Although simple, this definition hides a problem. To understand it better, let's look at an example. Let's try to find the GCD between the following monomials:

In looking for the GCD between these two monomials, it must be kept in mind that they must be divisible by the result. Therefore, it is necessary to remember the rule of divisibility between monomials.

This rule gives us a first clue as to what the final monomial has to be. In fact, should be divisible by the result and, therefore, the letters of this monomial shall appear in the result. Similarly, the monomial must also be divisible by the result and all its letters shall also appear.

It follows that the result must contain all the letters common to the given monomials.

Starting with the two monomials, we can easily see that the letters and must appear in the result. The letter cannot appear because it is not common to both.

As far as the exponents of the letters are concerned, we can always proceed from the definition of divisibility between monomials. For a monomial to be divisible by a second monomial, the letters which appear in the second monomial must have an exponent less than or equal to the exponent with which they appear in the first monomial.

It then follows that each letter of the result must have the minimum exponent with which that letter appears in the given monomials. This is why the letter must have the exponent 2 and the letter must have the exponent 1. The result could be the following monomial:

We can, in fact, easily see that this monomial simultaneously divides both and .

However, this monomial is not the only greatest common divisor. We could also have chosen the monomial , in fact, if we try to divide the two starting monomials by the latter we get:

In simple terms, what determines the Maximum Common Divisor between monomials is exclusively the literal part. As a result, we can choose any number as a numerical coefficient and, therefore, the GCDs are infinite.

However, rather than choosing any coefficient, when calculating the GCD of two or more monomials the following convention is followed:

• If one or more of the coefficients of the given monomials are not integers, the coefficient of the Maximum Common Divisor is set to 1.
• If, on the other hand, all the coefficients of the given monomials are integers, the coefficient of the Maximum Common Divisor is set equal to the Greatest Common Divisor of the absolute values of the coefficients.

Following this convention, in the example above we have the coefficients 7 and 5. They are integers, but their GCD is 1. Therefore, the final result will have the coefficient 1:

Definition of Greatest Common Divisor between Monomials

In view of the above considerations, we give the definition of the Greatest Common Divisor, or GCD, between monomials:

GCD between monomials Example

Let's look at another example. Let's take the following monomials and calculate their GCD:

First, let's calculate the coefficient of the GCD. As can be observed, all coefficients are integers, so the final coefficient will be the GCD of the absolute values of the coefficients:

Thus, the resulting monomial will have as coefficient the number 3.

Next, calculate the resulting literal part. To do this, a simple way is to line up all the letters present in the starting monomials, like this:

Next, we only take the letters that appear in all the rows:

Finally, we impose on the letters the smallest exponent with which they appear in their column:

So the end result will be:

Lowest Common Multiple between Monomials

We can define the Lowest Common Multiple between Monomials as an any monomial that is simultaneously divisible by all given monomials.

Again, we can use the same reasoning as in the case of the GCD.

Let's start with an example: we want to calculate the lcm of these two monomials:

Recalling the rule of divisibility between monomials, we should have that the final result must be divisible by both the monomials of the example. From this we can derive two pieces of information:

1. The result must simultaneously have all the letters of the first monomial, , and that of the second monomial . So the result must contain the three letters , and . Otherwise, it could not be divisible simultaneously by both monomials.
2. For the result to be divisible by both monomials, each letter must have the maximum exponent with which it appears in the two monomials. Thus, the letter must have the number 3 as its exponent, the letter must have the number 2 as its exponent and the letter must have the number 1 as its exponent.

The literal part of the final result will then be:

It remains to find the numerical coefficient. Again, as with the GCD, we can choose any numerical coefficient. That is, the lcm of two or more given monomials are infinite. This is why we follow the same convention for the case of the GCD. The numeric coefficient of the lcm of two or more monomials is:

• Equal to 1 if one or more monomials have a non-integer coefficient,
• Equal to the lcm of the absolute values of the numerical coefficients if all the coefficients of the given monomials are integer.

In the example above, and are both integers so we shall use the lcm of their absolute values which is 1. So the final result is:

Definition of Lowest Common Multiple between Monomials

Having taken the above considerations into account, we define the Lowest Common Multiple, or lcm, between monomials:

lcm between monomials Example

Let's look at the example from before. This time, however, we calculate the lcm of the given monomials:

First, let's calculate the coefficient of the lcm. As can be seen, all coefficients are integers, so the final coefficient will be the lcm of the absolute values of the coefficients:

Thus, the resulting monomial will have the number 12 as a coefficient.

We then calculate the resulting literal part. To do this, we use the same technique as for GCD, i.e. lining up all the letters present in the starting monomials, like this:

Next, take all letters, taken once per column:

Finally, we impose on the letters the largest exponent with which they appear in their column:

So the end result will be:

In brief

The aim of this lesson is to derive the general rule for calculating the GCD between monomials and lcm between monomials.

We have seen that, in practice, what determines the GCD and the lcm is the literal part of the monomials:

• In the case of GCD between monomials we take only the common letters, only once and with minimum exponent,
• In the case of lcm between monomials, we take all the letters that appear in at least one monomial, taken once and with a maximum exponent.

As far as the numerical coefficient is concerned, on the other hand, the following convention is followed:

• 1 is chosen as coefficient in case at least one of the coefficients of the given monomials is not an integer.
• We take, respectively, the GCD or the lcm of the absolute values of the coefficients if all the coefficients are all integers.

This lesson concludes the chapter on monomials. In the next chapter, we will begin the study of Polynomials.