Certainly! Let's solve this step-by-step.
Given:
- The Least Common Multiple (L.C.M) of two numbers is 120.
- The Highest Common Factor (H.C.F) of two numbers is 6.
- One of the numbers is 21.
We need to find the other number.
There is a fundamental relationship between L.C.M, H.C.F, and the two numbers, which states:
[tex]\[ \text{LCM} \times \text{HCF} = \text{num1} \times \text{num2} \][/tex]
In this problem, we are given:
[tex]\[ \text{LCM} = 120 \][/tex]
[tex]\[ \text{HCF} = 6 \][/tex]
[tex]\[ \text{num1} = 21 \][/tex]
We need to find [tex]\(\text{num2}\)[/tex].
Using the relationship:
[tex]\[ 120 \times 6 = 21 \times \text{num2} \][/tex]
First, calculate the product of LCM and HCF:
[tex]\[ 120 \times 6 = 720 \][/tex]
So we have:
[tex]\[ 720 = 21 \times \text{num2} \][/tex]
To find [tex]\(\text{num2}\)[/tex], divide both sides of the equation by 21:
[tex]\[ \text{num2} = \frac{720}{21} \][/tex]
When you perform this division, you get the result:
[tex]\[ \text{num2} = 34 \][/tex]
Therefore, the other number is:
[tex]\[ \boxed{34} \][/tex]
So the two numbers are 21 and 34, with an L.C.M of 120 and an H.C.F of 6.
