Question

HCF of two numbers \[ = \]
A. Product of numbers \[ + \] their LCM
B. Product of numbers \[ - \] their LCM
C. Product of numbers \[ \times \] their LCM
D. Product of numbers \[ \div \] their LCM
E. Answer required

Answer 1

Hint: We use the concept of least common multiple (LCM) and highest common factor (HCF) of two numbers to write the value of
* LCM: LCM means least common multiple is the smallest number that is multiple of two or more given numbers.
* HCF of two numbers is the highest common factor that divides both the numbers.

Complete step-by-step solution:
Here we take the case of two numbers
Since we know that LCM\[ \times \]HCF\[ = \]Product of two numbers …………...… (1)
If we divide both sides of the equation (1) by LCM of the two numbers
\[ \Rightarrow \]LCM\[ \times \]HCF\[ \div \]LCM\[ = \]Product of two numbers\[ \div \]LCM
Cancel same terms i.e. LCM from numerator and denominator in left hand side of the equation
\[ \Rightarrow \]HCF \[ = \]Product of two numbers\[ \div \]LCM

\[\therefore \]Option D is correct.

Note: Alternate method:
We can show this solution using an example
Let us take two numbers as 4 and 6
Product of two numbers is \[4 \times 6 = 24\].................… (1)
Now we write the prime factorization of both the numbers and calculate LCM and HCF
\[ \Rightarrow \]Prime factorization of \[4 = 2 \times 2\] and \[6 = 2 \times 3\]
Since LCM is the least common multiple of two numbers; LCM of 4 and 6 is \[2 \times 2 \times 3 = 12\]
And HCF is the highest common factor of two numbers; HCF of 4 and 6 is 2
\[ \Rightarrow \]LCM\[ = 12\] and HCF\[ = 2\]
Now we calculate the value of HCF of two numbers using the value of LCM and product of the two numbers.
Since HCF\[ = 2\]
We can write \[2 = \dfrac{{24}}{{12}}\]
Substitute the value of 24 as product of two numbers and 12 as LCM of two numbers and 2 as HCF of two numbers
\[ \Rightarrow \]HCF of two numbers\[ = \]Product of two numbers \[ \div \]LCM of two numbers
 \[\therefore \]Option D is correct