Question

The ratio of two numbers is \[5:6\] and their LCM is \[480\], then their HCF is:
A. \[20\]
B. \[16\]
C. \[6\]
D. \[5\]

Answer 1

Hint: Here we will be using the formula of calculating LCM when two numbers and their HCF is given. The formula is as below:
\[{\text{L}}{\text{.C}}{\text{.M}} \times {\text{H}}{\text{.C}}{\text{.F}} = \left( {x \times y} \right)\] , where \[x\] and \[y\] are two numbers.

Complete step-by-step solution:
Step 1: It is given that the ratio of two numbers is \[5:6\] and their LCM is \[480\].
Let the two numbers are \[{\text{5}}a\] and \[6a\]. So, their HCF will be equal to \[a\].
By using the formula of calculating HCF we get:
\[ \Rightarrow {\text{L}}{\text{.C}}{\text{.M}} \times a = \left( {5a \times 6a} \right)\]
Step 2: By substituting the value of LCM in the above expression \[{\text{L}}{\text{.C}}{\text{.M}} \times a = \left( {5a \times 6a} \right)\], we get:
\[ \Rightarrow 480 \times a = \left( {5a \times 6a} \right)\]
We can write the above expression as below:
\[ \Rightarrow 480 \times a = 5a \times 6a\]
Step 3: By multiplying the terms in the RHS side of the above expression \[480 \times a = 5a \times 6a\], we get:
\[ \Rightarrow 480a = 30{a^2}\]
By bringing \[30a\] into the LHS side of the above expression
\[480a = 30{a^2}\], we get:
\[ \Rightarrow \dfrac{{480a}}{{30a}} = a\]
By doing the final division, we get:
\[ \Rightarrow 16 = a\]
So, the value of HCF is \[16\].

Option B is correct.

Note: Students need to remember the formula for calculating LCM/HCF when one of them is given with the two numbers.
Also, you should understand the difference between LCM and HCF. LCM is known as the least common multiple of any two or more natural numbers and HCF is known as the largest or greatest common factor to any two or more given numbers. Some properties of them are given below:
HCF of co-prime numbers is \[1\]. Therefore, the LCM of given co-prime numbers is equal to the product of those numbers.
HCF and LCM of the fractions are equals to as below:
\[{\text{LCM of fractions}} = \dfrac{{{\text{LCM }}{\text{of numerator}}}}{{{\text{HCF of denominators}}}}\] , \[{\text{HCF of fractions}} = \dfrac{{{\text{HCF }}{\text{of numerator}}}}{{{\text{LCM of denominators}}}}\]