# LCM of monomials using factorization method

In this chapter we will learn to find LCM of given monomials using factorization method.

While calculating the LCM we are trying to find the lowest algebraic expression which gets fully divided by all the given monomials.

## Monomial LCM -Using Factorization method

To find the LCM of given monomials, follow the below steps;

(a) Break each of the monomial into individual factors.

(b) Compare the factors of all monomials.

After comparison, you have to select each factors with the highest power.

(c) Write all the selected components together and you will get the LCM.

I hope you understood the above three steps.

Let us now solve some questions for better clarity.

## Finding LCM of monomials – Solved Examples

Example 01
Find the LCM of below monomials

\mathtt{\Longrightarrow \ 22xy^{3} \ }\\\ \\ \mathtt{\Longrightarrow \ 8x^{2} y^{2}}\

Solution
To find the LCM, follow the below steps;

(a) Factorize the monomial into smaller components.

\mathtt{22xy^{3} \Longrightarrow \ 2\times 11\times \ x\ \times \ y^{3} \ }\\\ \\ \mathtt{8x^{2} y^{2} \Longrightarrow \ 2^{3} \times x^{2} \times y^{2}}

(b) Compare the factors and select each factor with highest power.

Highest power of factor 2 ⟹ 3
Highest power of factor 11 ⟹ 1
Highest power of factor x ⟹ 2
Highest power of factor y ⟹ 3

Writing all the highest factors together;

\mathtt{\Longrightarrow 2^{3} \times 11\times x^{2} \times y^{3}}\\\ \\ \mathtt{\Longrightarrow \ 8\ \times 11\times x^{2} \times y^{3}}\\\ \\ \mathtt{\Longrightarrow \ 88x^{2} y^{3}}

Hence, \mathtt{88x^{2} y^{3}} is the LCM of given monomials.

Example 02
Find LCM of given monomials

\mathtt{\Longrightarrow \ 9x^{5} yz^{3}}\\\ \\ \mathtt{\Longrightarrow \ 12x^{3} y^{2} z^{2}}

Solution
Follow the below steps;

(a) Factorize the monomial into smaller components

\mathtt{9x^{5} yz^{3} \Longrightarrow \ 3^{2} \times x^{5} \times y\times z^{3}}\\\ \\ \mathtt{12x^{3} y^{2} z^{2} \Longrightarrow 2^{2} \times 3\times \ x^{3} \times y^{2} \times z^{2}}

(b) Compare the factors and select the ones with highest power.

Highest power of factor 2 ⟹ 2
Highest power of factor 3 ⟹ 2
Highest power of factor x ⟹ 5
Highest power of factor y ⟹ 2
Highest power of factor z ⟹ 3

Combining all the highest factors;

\mathtt{\Longrightarrow \ 2^{2} \times 3^{2} \times x^{5} \times y^{2} \times z^{3}}\\\ \\ \mathtt{\Longrightarrow \ 4\times 9\times x^{5} y^{2} z^{3}}\\\ \\ \mathtt{\Longrightarrow \ 36\ x^{5} y^{2} z^{3}}

Hence, \mathtt{36\ x^{5} y^{2} z^{3}} is the LCM of given monomials.

Example 03
Find LCM of given monomials.

\mathtt{\Longrightarrow \ 5x^{2} y}\\\ \\ \mathtt{\Longrightarrow \ 30y^{7}}\\\ \\ \mathtt{\Longrightarrow \ 45xy^{2}}

Solution
Follow the below steps;

(a) Factorize each monomial into smaller components.

\mathtt{5x^{2} y\Longrightarrow \ 5\times x^{2} \times y}\\\ \\ \mathtt{30y^{7} \Longrightarrow \ 2\times 3\times 5\times y^{7}}\\\ \\ \mathtt{45xy^{2} \ \Longrightarrow \ 3^{2} \times 5\times x\times y^{2}}

(b) Compare each factors and select the one with highest power.

Highest power of factor 2 ⟹ 1
Highest power of factor 3 ⟹ 2
Highest power of factor 5 ⟹ 1
Highest power of factor x ⟹ 2
Highest power of factor y ⟹ 7

Combining all the highest factors;

\mathtt{\Longrightarrow \ 2\times 3^{2} \times 5\times x^{2} \times y^{7}}\\\ \\ \mathtt{\Longrightarrow \ 2\times 9\times 5\times x^{2} y^{7}}\\\ \\ \mathtt{\Longrightarrow \ 90x^{2} y^{7}}

Hence, \mathtt{90x^{2} y^{7}} is the LCM of given monomials.

Example 04
Find the LCM of below monomials.

\mathtt{\Longrightarrow 18x^{2} y^{3} z^{4}}\\\ \\ \mathtt{\Longrightarrow 9xy^{4} z^{3}}

Solution
Follow the below steps;

(a) Factorize each monomial into smaller components.

\mathtt{18x^{2} y^{3} z^{4} \Longrightarrow 2\times 3^{2} \times x^{2} \times y^{3} \times z^{4}}\\\ \\ \mathtt{9xy^{4} z^{3} \Longrightarrow 3^{2} \times x\times y^{4} \times z^{3}}

(b) Compare each factors and select the one with highest power.

Highest power of factor 2 ⟹ 1
Highest power of factor 3 ⟹ 2
Highest power of factor x ⟹ 2
Highest power of factor y ⟹ 4
Highest power of factor z ⟹ 4

Combining all the highest factors;

\mathtt{\Longrightarrow \ 2\times 3^{2} \times x^{2} \times y^{4} \times z^{4}}\\\ \\ \mathtt{\Longrightarrow \ 2\times 9\times x^{2} y^{4} z^{4}}\\\ \\ \mathtt{\Longrightarrow \ 18x^{2} y^{4} z^{4}}

Hence, \mathtt{18x^{2} y^{4} z^{4}} is the LCM of given monomial.

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