Least Common Multiple of Monomials
How to determine the least common multiple of monomials
The least common multiple (LCM) of two or more monomials is a monomial whose coefficient is the LCM of the absolute values of the numerical coefficients, and whose literal part consists of every variable that appears in any of the monomials, each raised to its highest exponent across the set.
When referring to the least common multiple (LCM) of two or more monomials, we mean the monomial of lowest possible degree that is divisible by all the given monomials.
To determine it, we apply two simple rules:
- Literal part: include all variables that appear in any of the monomials - both shared and distinct - using each variable only once, and assign to it the highest exponent with which it appears in any of the monomials.
- Numerical coefficient: if the coefficients are integers, take their least common multiple. If they are fractional or decimal, the numerical coefficient of the LCM is conventionally taken to be 1. This is because, in such cases, attention is primarily given to the literal part of the monomial.
A worked example
Let’s examine the following monomials:
$$ 8 a^2b^3c $$
$$ 4 a^4b^2 $$
$$ 6 a^3b^4c^2 $$
We begin by finding the least common multiple of the absolute values of the numerical coefficients.
In this case, the LCM of 8, 4, and 6 is:
$$ \text{lcm}(8, 4, 6) = 2^3 \cdot 3 = 24 $$
Note. If the coefficients are not all integers, we simply take 1 as the numerical coefficient for the LCM.
Next, we align the literal parts (i.e., the variables and their exponents):
$$ \begin{array}{|clcc} a^2 & b^3 & c \\ a^4 & b^2 & \\ a^3 & b^4 & c^2 \\ \hline \\ a^4 & b^4 & c^2 \end{array} $$
We now take each variable that appears (a, b, and c), select the highest exponent it has among all monomials, and form their product:
$$ a^4 \cdot b^4 \cdot c^2 $$
Finally, we multiply this literal part by the LCM of the numerical coefficients:
Thus, the least common multiple of the three monomials is: 24·a4b4c2
$$ 24a^4b^4c^2 $$
Example 2
Now, let’s determine the LCM of the following monomials:
$$ \frac{3}{5} a^4b^5 $$
$$ 8 b^3c^2 $$
$$ 2 ab^2c $$
Since not all coefficients are integers, we treat the numerical part as 1 for the purposes of simplification:
$$ \text{lcm}\left( \frac{3}{5}, 8, 2 \right) = 1 $$
Note. This is a simplification often used to avoid unnecessarily complex calculations. In practice, the literal part is typically more relevant in such operations. Alternatively, we could replace only the fractional coefficient with 1 and compute the LCM of the remaining integers: $$ \text{lcm}(1, 8, 2) = 8 $$ Or, for greater precision, convert all coefficients into fractions and compute: $$ \text{lcm}\left( \frac{3}{5}, \frac{8}{1}, \frac{2}{1} \right) = \frac{\text{lcm}(3,8,2)}{\gcd(5,1,1)} = \frac{24}{1} = 24 $$
We now align the literal parts and proceed as before, selecting each variable only once and assigning it its highest exponent:
$$ \begin{array}{|clcc} a^4 & b^5 & \\ & b^3 & c^2 \\ a & b^2 & c \\ \hline \\ a^4 & b^5 & c^2 \end{array} $$
So, the least common multiple of the three monomials is:
$$ \text{lcm} = a^4b^5c^2 $$
And so on.
