Explanations:The problem involves finding the Least Common Multiple (LCM) of two numbers, 72 and 120, given that their Highest Common Factor (HCF) is 24. The relationship between the HCF and LCM of two numbers is a fundamental concept in number theory and is given by the formula:
\[ \text{HCF} \times \text{LCM} = \text{Product of the numbers} \]
Let's break down the steps to understand this relationship and how it applies to the given problem:
1. **Understanding HCF and LCM:**
- The HCF (or GCD) of two numbers is the largest number that divides both of them without leaving a remainder.
- The LCM of two numbers is the smallest number that is a multiple of both of them.
2. **Given Information:**
- HCF of 72 and 120 is 24.
- We need to find the LCM of 72 and 120.
3. **Using the Relationship:**
- According to the relationship between HCF and LCM, we have:
\[ \text{HCF} \times \text{LCM} = \text{Product of the numbers} \]
- Substituting the given values:
\[ 24 \times \text{LCM} = 72 \times 120 \]
4. **Solving for LCM:**
- First, calculate the product of the numbers:
\[ 72 \times 120 = 8640 \]
- Now, solve for LCM:
\[ 24 \times \text{LCM} = 8640 \]
\[ \text{LCM} = \frac{8640}{24} \]
\[ \text{LCM} = 360 \]
5. **Conclusion:**
- The LCM of 72 and 120 is 360, which satisfies the relationship HCF × LCM = Product of the numbers.
By understanding the relationship between HCF and LCM, we can efficiently solve problems involving these concepts. This relationship is particularly useful in competitive exams and other mathematical contexts where quick calculations are necessary.