The HCF and LCM of two numbers are 12 and 72 respectively. If the sum of the two numbers is 60, then one of the numbers will be

Options

A. 12
B. 24
C. 60
D. 72

Answer

To solve the problem step-by-step, we will use the properties of HCF (Highest Common Factor) and LCM (Lowest Common Multiple) along with the given information.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- HCF of two numbers = 12
- LCM of two numbers = 72
- Sum of the two numbers = 60

2. **Let the Two Numbers be x and y**:
- We can denote the first number as \( x \) and the second number as \( y \).
- According to the problem, we have:
\[ x + y = 60 \]
- From this equation, we can express \( y \) in terms of \( x \):
\[ y = 60 - x \]

3. **Use the Relationship Between HCF, LCM, and the Product of the Numbers**:
- The relationship states that:
\[ \text{HCF} \times \text{LCM} = x \times y \]
- Substituting the values we have:
\[ 12 \times 72 = x \times (60 - x) \]
- Calculate the left side:
\[ 864 = x(60 - x) \]

4. **Rearranging the Equation**:
- Expanding the right side gives:
\[ 864 = 60x - x^2 \]
- Rearranging this equation leads to:
\[ x^2 - 60x + 864 = 0 \]

5. **Solving the Quadratic Equation**:
- We can solve the quadratic equation \( x^2 - 60x + 864 = 0 \) using the factorization method.
- We need to find two numbers that multiply to 864 and add up to 60. These numbers are 36 and 24.
- Therefore, we can factor the equation as:
\[ (x - 36)(x - 24) = 0 \]
- This gives us the solutions:
\[ x = 36 \quad \text{or} \quad x = 24 \]

6. **Finding the Two Numbers**:
- If \( x = 36 \), then:
\[ y = 60 - 36 = 24 \]
- If \( x = 24 \), then:
\[ y = 60 - 24 = 36 \]
- Thus, the two numbers are 36 and 24.

7. **Conclusion**:
- One of the numbers is \( 24 \).