If the difference between an exterior angle of a regular polygon of 'n' sides and an exterior angle of another regular polygon of `'(n+1)'` sides is equal to `5^(@)`, find the value of 'n'.

Answer

To solve the problem, we need to find the value of 'n' such that the difference between the exterior angle of a regular polygon with 'n' sides and that of a regular polygon with '(n+1)' sides is equal to 5 degrees.

### Step-by-Step Solution:

1. **Understand the formula for the exterior angle of a polygon**:
The exterior angle of a regular polygon with 'n' sides is given by the formula:
\[ \text{Exterior angle} = \frac{360}{n} \]
Similarly, for a polygon with '(n+1)' sides, the exterior angle is:
\[ \text{Exterior angle} = \frac{360}{n+1} \]

2. **Set up the equation for the difference**:
According to the problem, the difference between the exterior angles of the two polygons is 5 degrees:
\[ \frac{360}{n} - \frac{360}{n+1} = 5 \]

3. **Simplify the equation**:
To simplify the left-hand side, we can find a common denominator:
\[ \frac{360(n+1) - 360n}{n(n+1)} = 5 \]
This simplifies to:
\[ \frac{360}{n(n+1)} = 5 \]

4. **Cross-multiply to eliminate the fraction**:
Cross-multiplying gives us:
\[ 360 = 5n(n+1) \]

5. **Rearranging the equation**:
Dividing both sides by 5:
\[ 72 = n(n+1) \]

6. **Form a quadratic equation**:
Rearranging gives:
\[ n^2 + n - 72 = 0 \]

7. **Factor the quadratic equation**:
We need to factor the quadratic equation. We look for two numbers that multiply to -72 and add to 1. The numbers are 9 and -8:
\[ (n - 8)(n + 9) = 0 \]

8. **Solve for 'n'**:
Setting each factor to zero gives:
\[ n - 8 = 0 \quad \Rightarrow \quad n = 8 \]
\[ n + 9 = 0 \quad \Rightarrow \quad n = -9 \quad (\text{not valid since } n \text{ must be positive}) \]

9. **Conclusion**:
Therefore, the value of 'n' is:
\[ n = 8 \]