Find the sum of all integers $n$ not less than $3$ such that the measure, in degrees, of an interior angle of a regular $n$-gon is an integer.

[i]2016 CCA Math Bonanza Team #3[/i]
To solve the problem, we need to find the sum of all integers \( n \) not less than 3 such that the measure, in degrees, of an interior angle of a regular \( n \)-gon is an integer.

1. **Formula for the Interior Angle:**
   The measure of an interior angle of a regular \( n \)-gon is given by:
   \[
   \theta = \frac{180(n-2)}{n}
   \]
   We need this to be an integer.

2. **Condition for Integer Interior Angle:**
   For \(\theta\) to be an integer, \(\frac{180(n-2)}{n}\) must be an integer. This implies that \(180(n-2)\) must be divisible by \(n\).

3. **Simplifying the Divisibility Condition:**
   \[
   \frac{180(n-2)}{n} = 180 - \frac{360}{n}
   \]
   For this to be an integer, \(\frac{360}{n}\) must be an integer. Therefore, \(n\) must be a divisor of 360.

4. **Finding Divisors of 360:**
   The prime factorization of 360 is:
   \[
   360 = 2^3 \times 3^2 \times 5
   \]
   The number of divisors is given by:
   \[
   (3+1)(2+1)(1+1) = 4 \times 3 \times 2 = 24
   \]
   Listing the divisors of 360: \(1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360\).

5. **Filtering Divisors Greater than or Equal to 3:**
   We exclude 1 and 2, and consider the remaining divisors:
   \[
   3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
   \]

6. **Summing the Valid Divisors:**
   \[
   3 + 4 + 5 + 6 + 8 + 9 + 10 + 12 + 15 + 18 + 20 + 24 + 30 + 36 + 40 + 45 + 60 + 72 + 90 + 120 + 180 + 360
   \]

7. **Calculating the Sum:**
   \[
   3 + 4 + 5 + 6 + 8 + 9 + 10 + 12 + 15 + 18 + 20 + 24 + 30 + 36 + 40 + 45 + 60 + 72 + 90 + 120 + 180 + 360 = 1167
   \]

The final answer is \(\boxed{1167}\)