Greatest Common Divisor of Monomials

How to determine the greatest common divisor of monomials

The greatest common divisor (GCD) of two or more monomials is a monomial whose coefficient is the GCD of the absolute values of the numerical coefficients, and whose literal part consists of the variables shared by all monomials, each taken once with the smallest exponent among them.

In other words, given two or more monomials, the greatest common divisor (GCD) is the highest-degree monomial that divides all of them exactly.

• The literal part of the GCD is formed by taking the variables common to all the monomials, each raised to the smallest exponent with which it appears.
• The numerical coefficient is, by convention, the greatest common divisor of the numerical coefficients - provided they are integers. If the coefficients are not integers, the GCD is conventionally taken to be 1.

  For instance, if the coefficients are fractional, we generally take the numerical part of the GCD to be 1, since divisibility among monomials typically refers only to their literal (variable) components.

A step-by-step example

Let’s consider the following monomials:

$$ 8 a^2b^3c $$

$$ 4 a^4b^2 $$

$$ 6 a^3b^4c^2 $$

We start by finding the greatest common divisor of the numerical coefficients.

In this case, the GCD of 8, 4, and 6 is:

$$ GCD(8, 4, 6) = 2 $$

Note. If the numerical coefficients have no common divisor other than 1 - that is, if they’re relatively prime - the GCD is simply 1.

Next, we arrange the variables from each monomial in columns to identify those they have in common:

$$ \begin{array}{|clcc} \color{red}{a^2} & b^3 & c \\ a^4 & \color{red}{b^2} & \\ a^3 & b^4 & c^2 \\ \hline \\ a^2 & b^2 & - \end{array} $$

We then select the variables that appear in every monomial (in this case, a and b) and use the smallest exponent for each. We multiply them together:

$$ a^2 \cdot b^2 $$

Finally, we combine this with the numerical GCD to get the overall result:

So, the greatest common divisor of the three monomials is:

$$ 2a^2b^2 $$

Example 2

Now let’s find the GCD of the following monomials:

$$ \frac{3}{5} a^4b^5 $$

$$ 8 b^3c^2 $$

$$ 2 ab^2c $$

Even though one of the coefficients is a fraction, the approach stays the same.

The numerical coefficients don’t share any common factor other than 1, so we take 1 as the numerical part of the GCD:

$$ GCD\left(\frac{3}{5}, 8, 2\right) = 1 $$

We now organize the variables, identify those that appear in all monomials (just b in this case), and use the lowest exponent:

$$ \begin{array}{|clcc} a^4 & b^5 & \\ & b^3 & c^2 \\ a & \color{red}{b^2} & c \\ \hline \\ - & b^2 & - \end{array} $$

Here, b is the only variable shared by all three.

So, the greatest common divisor of these monomials is:

$$ GCD = b^2 $$

And the method continues in the same way for any number of monomials.