Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers.
1. The measure of each interior angle of a regular $n$-gon can be calculated using the formula:
   \[
   \text{Interior angle} = \frac{(n-2) \cdot 180^\circ}{n}
   \]
   Simplifying this, we get:
   \[
   \text{Interior angle} = 180^\circ - \frac{360^\circ}{n}
   \]

2. For the interior angle to be an integer, $\frac{360^\circ}{n}$ must also be an integer. This means that $n$ must be a divisor of $360$.

3. To find the number of positive divisors of $360$, we first find its prime factorization:
   \[
   360 = 2^3 \cdot 3^2 \cdot 5
   \]

4. The number of divisors of a number is given by multiplying the incremented exponents of its prime factorization:
   \[
   (3+1)(2+1)(1+1) = 4 \cdot 3 \cdot 2 = 24
   \]

5. We exclude $n = 1$ and $n = 2$ because a 1-gon and a 2-gon do not exist as convex polygons. Therefore, we subtract these two cases from the total number of divisors:
   \[
   24 - 2 = 22
   \]

Conclusion:
\[
\boxed{22}
\]