The interior angle of a polygon are in A.P. with common difference `5^@`. If the smallest angle is `120^@`, find the number of sides of the polygon.

Answer

To solve the problem step by step, we will follow the logic used in the video transcript.

### Step 1: Understand the Problem
We are given that the interior angles of a polygon are in Arithmetic Progression (A.P.) with a common difference of \(5^\circ\) and the smallest angle is \(120^\circ\). We need to find the number of sides of the polygon.

### Step 2: Define Variables
Let:
- \(a = 120^\circ\) (the smallest angle)
- \(d = 5^\circ\) (the common difference)
- \(n\) = number of sides of the polygon

### Step 3: Sum of Interior Angles
The sum of the interior angles of a polygon with \(n\) sides is given by the formula:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]

### Step 4: Sum of Angles in A.P.
The angles in A.P. can be expressed as:
- First angle: \(a = 120^\circ\)
- Second angle: \(a + d = 120^\circ + 5^\circ = 125^\circ\)
- Third angle: \(a + 2d = 120^\circ + 10^\circ = 130^\circ\)
- ...
- Last angle: \(a + (n-1)d = 120^\circ + (n-1) \times 5^\circ\)

The sum of the first \(n\) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Substituting the values of \(a\) and \(d\):
\[ S_n = \frac{n}{2} \times (2 \times 120 + (n - 1) \times 5) \]
\[ S_n = \frac{n}{2} \times (240 + 5n - 5) \]
\[ S_n = \frac{n}{2} \times (5n + 235) \]

### Step 5: Equate the Two Sums
Now, we equate the sum of the interior angles to the sum of the angles in A.P.:
\[ \frac{n}{2} \times (5n + 235) = (n - 2) \times 180 \]

### Step 6: Simplify the Equation
Cross-multiplying gives:
\[ n(5n + 235) = 2(n - 2) \times 180 \]
\[ n(5n + 235) = 360n - 720 \]
Expanding the left side:
\[ 5n^2 + 235n = 360n - 720 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ 5n^2 + 235n - 360n + 720 = 0 \]
\[ 5n^2 - 125n + 720 = 0 \]

### Step 8: Divide by 5
Dividing the entire equation by 5:
\[ n^2 - 25n + 144 = 0 \]

### Step 9: Factor the Quadratic Equation
We need to factor the quadratic equation:
\[ n^2 - 25n + 144 = 0 \]
We look for two numbers that multiply to \(144\) and add to \(-25\). The numbers are \(-9\) and \(-16\):
\[ (n - 9)(n - 16) = 0 \]

### Step 10: Solve for \(n\)
Setting each factor to zero gives:
1. \(n - 9 = 0 \Rightarrow n = 9\)
2. \(n - 16 = 0 \Rightarrow n = 16\)

### Step 11: Check Validity
Since the interior angles of a polygon cannot exceed \(180^\circ\), we check:
- For \(n = 16\): The angles would be \(120^\circ, 125^\circ, \ldots, 195^\circ\) (which is not valid).
- For \(n = 9\): The angles are \(120^\circ, 125^\circ, \ldots, 160^\circ\) (which is valid).

### Conclusion
Thus, the number of sides of the polygon is:
\[ \boxed{9} \]