The ratio of two numbers is `7:13` and their HCF is 8. Their LCM is

Options

A. 728
B. 872
C. 628
D. 782

Answer

To find the LCM of two numbers given their ratio and HCF, we can follow these steps:

### Step 1: Understand the Ratio
The ratio of the two numbers is given as \(7:13\). This means we can represent the two numbers as:
- First number = \(7x\)
- Second number = \(13x\)

### Step 2: Use the HCF
We are given that the HCF of the two numbers is \(8\). Since the HCF of the numbers \(7x\) and \(13x\) is \(x\) (because 7 and 13 are coprime), we can set:
\[ x = 8 \]

### Step 3: Calculate the Actual Numbers
Now substituting \(x\) back into the expressions for the numbers:
- First number = \(7x = 7 \times 8 = 56\)
- Second number = \(13x = 13 \times 8 = 104\)

### Step 4: Use the Relationship Between HCF, LCM, and the Numbers
We know that:
\[ \text{HCF} \times \text{LCM} = \text{First Number} \times \text{Second Number} \]
Substituting the known values:
\[ 8 \times \text{LCM} = 56 \times 104 \]

### Step 5: Calculate the Product of the Numbers
Now calculate \(56 \times 104\):
\[ 56 \times 104 = 5824 \]

### Step 6: Solve for LCM
Now we can solve for LCM:
\[ 8 \times \text{LCM} = 5824 \]
\[ \text{LCM} = \frac{5824}{8} = 728 \]

### Final Answer
Thus, the LCM of the two numbers is \(728\).

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