L.C.M. of two numbers is 120 and their H.C.F. is 10 . Which of the following can be the sum of those two numbers

Options

A. 140
B. 89
C. 60
D. 70

Answer

To solve the problem, we need to find two numbers whose LCM is 120 and HCF is 10, and then determine which of the given options can be the sum of those two numbers.

### Step-by-Step Solution:

1. **Understanding the relationship between LCM and HCF**:
The relationship between the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two numbers can be expressed as:
\[ \text{LCM} \times \text{HCF} = \text{Product of the two numbers} \]
Given that LCM = 120 and HCF = 10, we can write:
\[ 120 \times 10 = \text{Product of the two numbers} \]
Therefore, the product of the two numbers is:
\[ 1200 \]

2. **Let the two numbers be \(10x\) and \(10y\)**:
Since the HCF is 10, we can express the two numbers as:
\[ \text{First number} = 10x \quad \text{and} \quad \text{Second number} = 10y \]
Here, \(x\) and \(y\) are coprime (i.e., their HCF is 1).

3. **Using the LCM formula**:
The LCM of the two numbers can also be expressed as:
\[ \text{LCM}(10x, 10y) = 10 \times \text{LCM}(x, y) \]
Since \(x\) and \(y\) are coprime, we have:
\[ \text{LCM}(x, y) = x \times y \]
Thus, we can write:
\[ \text{LCM}(10x, 10y) = 10xy \]

4. **Setting up the equation**:
From the LCM condition, we have:
\[ 10xy = 120 \]
Dividing both sides by 10 gives:
\[ xy = 12 \]

5. **Finding pairs of \(x\) and \(y\)**:
Now we need to find pairs of integers \(x\) and \(y\) such that their product is 12 and they are coprime:
- Possible pairs of factors of 12 are:
- (1, 12)
- (2, 6)
- (3, 4)

We check which pairs are coprime:
- (1, 12): Coprime
- (2, 6): Not coprime (common factor 2)
- (3, 4): Coprime

The valid pairs are (1, 12) and (3, 4).

6. **Calculating the sums**:
For each valid pair, we calculate the sum of the two numbers:
- For the pair (1, 12):
\[ \text{Sum} = 10 \times 1 + 10 \times 12 = 10 + 120 = 130 \]
- For the pair (3, 4):
\[ \text{Sum} = 10 \times 3 + 10 \times 4 = 30 + 40 = 70 \]

7. **Conclusion**:
The possible sums of the two numbers are 130 and 70. Therefore, the sum that can be the answer is:
\[ \boxed{70} \]