# LCM of monomials

In this chapter we will learn to find LCM of two or more monomials with solved examples.

At the end of the chapter some problems are also provided for practice.

To understand this chapter, you should have basic knowledge about the concept of Lowest common multiples and polynomials.

## Finding LCM of Monomials

Let us first review the basics.

### What are monomials?

The algebraic expression containing single terms are called monomials.

The term may contain constants, variables or both.

Given below are some examples of monomials;

\mathtt{( i) \ 4x^{2}}\\\ \\ \mathtt{( ii) \ 12xyz}\\\ \\ \mathtt{( iii) \ 45x^{3} y^{2} z}

### Significance of LCM of monomials

While calculating LCM of monomials, we are trying to find the lowest value which will get fully divided by the monomials.

Consider the two monomials \mathtt{4x^{2} \ and\ 3xy} .

The LCM of given monomial is \mathtt{12x^{2} y} .

Hence, \mathtt{12x^{2} y} is the lowest polynomial which will get fully divided by monomials \mathtt{4x^{2} \ and\ 3xy} .

### How to calculate LCM of monomials ?

Follow the below steps to calculate LCM of monomials;

(a) Separate the coefficients and variables.

(b) Find the LCM of coefficients separately.

(c) Separate the individual variables and select the ones which has the highest power.

(d) Join the LCM of coefficient and variables.

I hope you understand the above process. Let us see some solved examples

Example 01
Find LCM of monomials
\mathtt{\Longrightarrow \ 45x^{2} y^{3}}\\\ \\ \mathtt{\Longrightarrow \ 81xy^{5}}

Solution

(a) Find LCM of coefficients

LCM ( 45, 81) = 405

(b) Find highest power of variables

Variable x
Among given monomials, the highest power of x is 2

Variable y
The highest power of y variable is 5.

(c) Combine LCM of coefficients and variables.

\mathtt{LCM\ =\ 405\ x^{2} y^{5}}

Hence, \mathtt{405\ x^{2} y^{5}} is the lowest multiple which get fully divided by given monomials.

Example 02
Find LCM of below monomials

\mathtt{\Longrightarrow x^{4} y\ z^{2}}\\\ \\ \mathtt{\Longrightarrow \ 24y^{3} z^{3}}\\\ \\ \mathtt{\Longrightarrow \ 6xyz}

Solution
(a) Find LCM of coefficient

LCM (1, 24, 6) = 24

(b) Find highest power of all variables

Variable x
Highest power among given monomial is 4

Variable y
Highest power is 3

Variable z
Highest power is 3.

(c) Combine the LCM of coefficients and variables.

Hence, LCM of given monomial is \mathtt{\Longrightarrow \ 24x^{4} y^{3} z^{3}}

Example 03
Find the LCM of below monomials.

\mathtt{\Longrightarrow 35xy}\\\ \\ \mathtt{\Longrightarrow 20yz}\\\ \\ \mathtt{\Longrightarrow 10zx}

Solution

(a) Find LCM of coefficients.

LCM (35, 20, 10) = 140

(b) Find highest power of all variables

Variable x
Highest power of variable x among the given monomial is 1

Variable y
Highest power is 1

Variable z
Highest power is 1

(c) Combine the LCM of coefficient and variables.

Hence, the LCM of given monomial is 140xyz.

It means that 140xyz is the lowest multiple which will get divided by given monomials.

Example 04
Find the LCM of given monomials
\mathtt{\Longrightarrow 19\ m^{5} n^{6}}\\\ \\ \mathtt{\Longrightarrow 15\ m^{7} n^{4}}\\\ \\ \mathtt{\Longrightarrow 3\ m\ n}

Solution

(a) Find LCM of coefficients.

LCM (19, 15, 3) = 285

(b) Find highest power of all variables

Variable m
The highest power of variable m among given monomial is 7.

Variable n
Highest power is 6

(c) Combine LCM of coefficients and variables

Hence, the LCM of given monomial is \mathtt{285\ m^{7} n^{6}}

Example 05
Find the LCM of below monomials

\mathtt{\Longrightarrow 5\ x^{4} y^{2}}\\\ \\ \mathtt{\Longrightarrow 25\ x^{10} y}\\\ \\ \mathtt{\Longrightarrow 15\ x^{2} y^{15}}\\\ \\ \mathtt{\Longrightarrow \ 40\ x^{3} y{^{5}}}

Solution

(a) Find LCM of coefficients.

LCM ( 5, 25, 15, 40) = 600

(b) Find highest power of all variables

Variable x
Highest power of x is 10

Variable y
Highest power of y is 15

(c) Combining the LCM of coefficients and variables

\mathtt{600\ x^{10} y^{15}}

## Explanation of LCM calculation

I hope you understood the above process of calculating LCM of monomials.

Here we will understand the background of above process.

I will explain what we are actually doing and why we are doing it in order find LCM of monomials.

Consider two monomials \mathtt{4x^{2} y and\ 6x^{3} y^{2}}

Factorize the given monomial into smaller components.

\mathtt{4x^{2} y\Longrightarrow 2^{2} \times x^{2} \times y}\\\ \\ \mathtt{6x^{3} y^{2} \Longrightarrow \ 2\times 3\times x^{3} \times y^{2}}

Now among the given monomial, select each factor with highest power.

Highest power of 2 ⟹ 2
Highest power of 3 ⟹ 1
Highest power of x ⟹ 2
Highest power of y ⟹ 1

Combining all the terms we get;

\mathtt{\Longrightarrow \ 2^{2} .3.x^{2} .y}\\\ \\ \mathtt{\Longrightarrow \ 4.3.x^{2} y}\\\ \\ \mathtt{\Longrightarrow \ 12\ x^{2} y}

Hence, we got the LCM.