The H.C.F. and L.C.M. of two numbers are 6 and 840 respectively. If one of the number is 42, find the other number.

Options

A. `110`
B. `120`
C. `130`
D. None of these

Answer

To find the other number when the H.C.F. and L.C.M. of two numbers are given, we can use the relationship between these values. Here’s how to solve the problem step by step:

### Step 1: Understand the relationship
We know that the product of two numbers (let's call them N1 and N2) is equal to the product of their H.C.F. and L.C.M. This can be expressed as:
\[ N1 \times N2 = \text{H.C.F.} \times \text{L.C.M.} \]

### Step 2: Substitute the known values
From the problem, we know:
- H.C.F. = 6
- L.C.M. = 840
- One of the numbers (N1) = 42

We can substitute these values into the equation:
\[ 42 \times N2 = 6 \times 840 \]

### Step 3: Calculate the product of H.C.F. and L.C.M.
Now, calculate the right side of the equation:
\[ 6 \times 840 = 5040 \]

### Step 4: Set up the equation
Now we have:
\[ 42 \times N2 = 5040 \]

### Step 5: Solve for N2
To find N2, we can rearrange the equation:
\[ N2 = \frac{5040}{42} \]

### Step 6: Perform the division
Now, divide 5040 by 42:
\[ N2 = 120 \]

### Conclusion
The other number is **120**.

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