If M and N are two natural numbers, then `(LCM(M,N))/(HCF(M,N))` is ___________

Options

A. a natural number
B. a rational number which is not necessarily an integer
C. a real number which is not necessarily an integer
D. an irrational number.

Answer

To solve the problem, we need to find the value of \((LCM(M,N))/(HCF(M,N))\) where \(M\) and \(N\) are two natural numbers.

### Step-by-Step Solution:

1. **Understanding LCM and HCF**:
- The **LCM (Least Common Multiple)** of two numbers is the smallest number that is a multiple of both.
- The **HCF (Highest Common Factor)**, also known as GCD (Greatest Common Divisor), is the largest number that divides both numbers without leaving a remainder.

2. **Using the Relationship**:
- There is a well-known relationship between LCM and HCF of two numbers:
\[ LCM(M, N) \times HCF(M, N) = M \times N \]
- This means that if we know the LCM and HCF, we can find the product of the two numbers.

3. **Dividing LCM by HCF**:
- We can rearrange the relationship:
\[ \frac{LCM(M, N)}{HCF(M, N)} = \frac{M \times N}{(HCF(M, N))^2} \]
- This shows that dividing the LCM by the HCF gives us a value that depends on the product of the two numbers divided by the square of their HCF.

4. **Conclusion**:
- Since \(M\) and \(N\) are natural numbers, the result of \((LCM(M,N))/(HCF(M,N))\) will also be a natural number.
- More specifically, it will be equal to the product of the two numbers divided by the square of their HCF.

5. **Final Answer**:
- Therefore, \((LCM(M,N))/(HCF(M,N))\) is a natural number.