Question

The product of two numbers is \[5376\] , and their HCF is \[8\]. Then their LCM is ----------?
A. \[267\]
B. \[672\]
C. \[726\]
D. \[762\]

Answer 1

Hint: Here we will use the property of HCF and LCM which states that if there are two natural numbers \[x\] and \[y\]. Then the product of the two numbers will be equal to the product of their LCM and HCF.
\[{\text{L}}{\text{.C}}{\text{.M}}\left( {x,y} \right) \times {\text{H}}{\text{.C}}{\text{.F}}\left( {x,y} \right) = x \times y\]

Complete step-by-step solution:
Step 1: It is given that the product of any two numbers is \[5376\] , and their HCF is \[8\]. Suppose the two numbers are \[x\] and \[y\].
So, we can write as below:
\[xy = 5376\] and \[{\text{H}}{\text{.C}}{\text{.F}}\left( {x,y} \right) = 8\]
Step 2: By using the property of HCF and LCM as given below, we will find the value of their LCM.
\[{\text{L}}{\text{.C}}{\text{.M}}\left( {x,y} \right) \times {\text{H}}{\text{.C}}{\text{.F}}\left( {x,y} \right) = x \times y\] ….. (1)
By substituting the value of \[xy = 5376\] and \[{\text{H}}{\text{.C}}{\text{.F}}\left( {x,y} \right) = 8\] in the above equation (1), we get:
\[ \Rightarrow {\text{L}}{\text{.C}}{\text{.M}}\left( {x,y} \right) \times 8 = 5376\]
By bringing \[8\] into the RHS side of the equation \[{\text{L}}{\text{.C}}{\text{.M}}\left( {x,y} \right) \times 8 = 5376\], we get:
\[ \Rightarrow {\text{L}}{\text{.C}}{\text{.M}}\left( {x,y} \right) = \dfrac{{5376}}{8}\]
By dividing the \[5376\] within the above equation, we get:
\[ \Rightarrow {\text{L}}{\text{.C}}{\text{.M}}\left( {x,y} \right) = 672\]
Therefore, the LCM of the variables are \[672\].

Option B is the correct answer.

Note: Students need to remember the important properties of HCF and LCM of two variables which play an important role in solving these types of questions. The proof of the property which we have used in the above question is as below:
If \[a\] and \[b\] are two variables then, their product is equal to the product of their HCF and LCM:
\[{\text{L}}{\text{.C}}{\text{.M}}\left( {a,b} \right) \times {\text{H}}{\text{.C}}{\text{.F}}\left( {a,b} \right) = a \times b\]
For example, let \[3\] and \[8\] are two variables. Then their product will be equals to as below:
\[ \Rightarrow 3 \times 8 = 24\]
HCF of these two numbers will be equals to as below:
\[{\text{HCF}}\left( {3,8} \right) = 1\]
LCM of these two numbers will be equals to as below:
\[{\text{LCM}}\left( {3,8} \right) = 24\]
From here, we can see that the product of the numbers and product of their HCF and LCM is equal.