The resistance of the series combination of two resistances is S. When they are joined in parallel the total resistance is P. If S= nP then the minimum possible value of n is

Answer

To solve the problem, we need to find the minimum possible value of \( n \) given the relationship between the series and parallel combinations of two resistances \( R_1 \) and \( R_2 \).

### Step-by-Step Solution:

1. **Define the Resistances**:
Let the two resistances be \( R_1 \) and \( R_2 \).

2. **Calculate Series Resistance (S)**:
The total resistance when the resistances are in series is given by:
\[ S = R_1 + R_2 \]

3. **Calculate Parallel Resistance (P)**:
The total resistance when the resistances are in parallel is given by:
\[ P = \frac{R_1 R_2}{R_1 + R_2} \]

4. **Given Relationship**:
We know from the problem statement that:
\[ S = nP \]
Substituting the expressions for \( S \) and \( P \) gives:
\[ R_1 + R_2 = n \left( \frac{R_1 R_2}{R_1 + R_2} \right) \]

5. **Rearranging the Equation**:
Rearranging the above equation leads to:
\[ (R_1 + R_2)^2 = n R_1 R_2 \]
This can be rewritten as:
\[ n = \frac{(R_1 + R_2)^2}{R_1 R_2} \]

6. **Minimizing n**:
To find the minimum value of \( n \), we can use the fact that \( R_1 \) and \( R_2 \) should be equal for the minimum value. Let \( R_1 = R_2 = R \):
\[ n = \frac{(R + R)^2}{R \cdot R} = \frac{(2R)^2}{R^2} = \frac{4R^2}{R^2} = 4 \]

7. **Conclusion**:
Therefore, the minimum possible value of \( n \) is:
\[ n = 4 \]

### Final Answer:
The minimum possible value of \( n \) is \( 4 \).