In the following questions, each question has four options (A), (B), (C) and (D). Choose the correct option and indicate your correct response.
The H.C.F. and L.C.M. of two numbers are 21 and 420 respectively, one of these number is 84. Find the another number?

Options

A. 105
B. 126
C. 63
D. None of these

Answer

To find the second number given that the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) of two numbers are 21 and 420 respectively, and one of the numbers is 84, we can use the relationship between H.C.F., L.C.M., and the two numbers.

### Step-by-Step Solution:

1. **Understand the relationship**: The relationship between the H.C.F., L.C.M., and the two numbers can be expressed as:
\[ \text{H.C.F.} \times \text{L.C.M.} = \text{First Number} \times \text{Second Number} \]

2. **Substitute the known values**: Here, we know:
- H.C.F. = 21
- L.C.M. = 420
- First Number = 84
- Second Number = \( x \) (which we need to find)

Plugging these values into the formula gives:
\[ 21 \times 420 = 84 \times x \]

3. **Calculate the left side**: First, calculate \( 21 \times 420 \):
\[ 21 \times 420 = 8820 \]

4. **Set up the equation**: Now we have:
\[ 8820 = 84 \times x \]

5. **Solve for \( x \)**: To find \( x \), divide both sides by 84:
\[ x = \frac{8820}{84} \]

6. **Perform the division**: Now, calculate \( \frac{8820}{84} \):
\[ x = 105 \]

7. **Conclusion**: The second number is \( 105 \).

### Final Answer:
The second number is **105**.