Find the number of positive integers $n$ such that a regular polygon with $n$ sides has internal angles with measures equal to an integer number of degrees.
1. The measure of an internal angle of a regular polygon with \( n \) sides is given by:
   \[
   \text{Internal angle} = 180^\circ - \frac{360^\circ}{n}
   \]
   For this angle to be an integer, \(\frac{360^\circ}{n}\) must also be an integer.

2. This implies that \( n \) must be a divisor of \( 360 \). We need to find all positive integers \( n \) such that \( n \) is a divisor of \( 360 \).

3. First, we find the prime factorization of \( 360 \):
   \[
   360 = 2^3 \cdot 3^2 \cdot 5
   \]

4. The number of divisors of \( 360 \) can be calculated using the formula for the number of divisors of a number \( n = p_1^{e_1} \cdot p_2^{e_2} \cdots p_k^{e_k} \), which is:
   \[
   (e_1 + 1)(e_2 + 1) \cdots (e_k + 1)
   \]
   For \( 360 \):
   \[
   (3+1)(2+1)(1+1) = 4 \cdot 3 \cdot 2 = 24
   \]
   Therefore, \( 360 \) has 24 divisors.

5. However, we need to exclude \( n = 1 \) and \( n = 2 \) because a polygon with 1 or 2 sides does not exist in the context of regular polygons. 

6. Subtracting these two cases from the total number of divisors, we get:
   \[
   24 - 2 = 22
   \]

Conclusion:
The number of positive integers \( n \) such that a regular polygon with \( n \) sides has internal angles with measures equal to an integer number of degrees is \( \boxed{22} \).