Question

If HCF of (a, 8) is equal to 4 and LCM of (a, 8) is equal to 24, then determine the value of a, from the following options:
$
  {\text{A}}{\text{. 8}} \\
  {\text{B}}{\text{. 10}} \\
  {\text{C}}{\text{. 12}} \\
  {\text{D}}{\text{. 14}} \\
$

Answer 1

Hint: In order to find the value of a, we understand the definitions of LCM and HCF of a number and try to establish a relation between LCM and HCF and the numbers. We use the property of LCM and HCF of two numbers to solve for the answer.

Complete step-by-step answer:
Given Data,
HCF of (a, 8) = 4
LCM of (a, 8) = 24

The HCF of two or more numbers is nothing but the highest common factor of the given number, in other words it is the highest number by which all the given numbers can be divided.
The LCM of two or more numbers is defined as the least common multiple of the given set of numbers, or the least number which is the multiple of all the given numbers.

We use one of the property of LCM and HCF, which states that
The product of LCM and HCF of any two given natural numbers is equal to the product of the two numbers itself.
This property is only applicable to only two numbers.
Using this property, we can write
The product of a and 8 = the product of LCM and HCF of a and 8.
⟹a × 8 = 4 × 24
$
   \Rightarrow {\text{a = }}\dfrac{{4 \times 24}}{8} \\
   \Rightarrow {\text{a = 4}} \times 3{\text{ = 12}} \\
$
Therefore the value of the number a, is equal to 12.
Option C is the correct answer.

Note:In order to solve this type of questions the key is to know the concept of LCM and HCF of natural numbers and their properties.
Some other properties of LCM and HCF of numbers are as follows:
HCF of co-prime numbers is equal to 1. Therefore the LCM of co-prime numbers is equal to their product.
LCM of fractions =$\dfrac{{{\text{LCM of numerators}}}}{{{\text{HCF of denominators}}}}$
HCF of fractions =$\dfrac{{{\text{HCF of numerators}}}}{{{\text{LCM of denominators}}}}$
The HCF of a given set of numbers is never greater than any of the given numbers.
LCM of a given set of numbers is never lesser than any of the given numbers.