Question

The product of two numbers is $882$ and their LCM is $126$. What is the HCF of these numbers?

Answer 1

Hint: The given question revolves around the properties of HCF and LCM. We must know the result that the product of HCF and LCM of two numbers is equal to the product of the two numbers. So, we will substitute the values of LCM and the product of two numbers in order to get to the value of HCF of the two numbers.

Complete step-by-step answer:
So, we have the product of two numbers as $882$.
Also, we are provided with the least common multiple of the two numbers as $126$.
Now, we have to find the highest common factor of these two numbers.
So, we know that the product of highest common factor and least common multiple of two numbers is equal to the product of two numbers.
So, we have, ${\text{Product}}\,{\text{of}}\,{\text{two}}\,{\text{numbers}} = HCF \times LCM$
Now, substituting the values of the product of two numbers and LCM of the two numbers in the above formula, we get,
$ \Rightarrow {\text{882}} = HCF \times 126$
Now, we use the method of transposition to find the value of HCF from the above equation. We divide both sides of the equation by the number $126$.
\[ \Rightarrow HCF = \dfrac{{{\text{882}}}}{{126}}\]
Simplifying the calculations, we get,
\[ \Rightarrow HCF = 7\]
So, the highest common factor of the two numbers whose product is equal to $882$ and LCM is equal to $126$ is $7$.
So, the correct answer is “7”.

Note: There are various methods for finding the least common multiple and highest common factor of two given numbers. The simplest method to find LCM and HCF is by prime factorization method. Least common multiple is a product of common factors with highest power and all other non-common factors. HCF is the product of the lowest powers of all the common factors.