The L.C.M.of two numbers is 63 times their H.C.F. If one of the numbers is 35 and the sum of H.C.F and L.C.M is 320, then the sum of two numbers is

Options

A. 45
B. 150
C. 80
D. 310

Answer

To solve the problem step by step, we can follow these calculations:

1. **Understanding the relationship between LCM and HCF**:
The problem states that the LCM of two numbers is 63 times their HCF.
Let the HCF be denoted as \( h \). Therefore, we can express the LCM as:
\[ \text{LCM} = 63h \]

2. **Using the sum of LCM and HCF**:
We are also given that the sum of the LCM and HCF is 320:
\[ \text{LCM} + \text{HCF} = 320 \]
Substituting the expression for LCM from the first step:
\[ 63h + h = 320 \]
Simplifying this, we get:
\[ 64h = 320 \]

3. **Finding the value of HCF**:
To find \( h \), we divide both sides by 64:
\[ h = \frac{320}{64} = 5 \]

4. **Finding the LCM**:
Now that we have \( h \), we can find the LCM:
\[ \text{LCM} = 63h = 63 \times 5 = 315 \]

5. **Using the relationship between the numbers**:
We know one of the numbers is 35. Let the other number be \( x \). The product of the two numbers can be expressed in terms of LCM and HCF:
\[ \text{LCM} \times \text{HCF} = 35 \times x \]
Substituting the values of LCM and HCF:
\[ 315 \times 5 = 35 \times x \]
Simplifying this gives:
\[ 1575 = 35x \]

6. **Finding the value of the other number \( x \)**:
Dividing both sides by 35:
\[ x = \frac{1575}{35} = 45 \]

7. **Finding the sum of the two numbers**:
Now we can find the sum of the two numbers:
\[ 35 + 45 = 80 \]

Thus, the sum of the two numbers is **80**.