Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular
$n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along
the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum
possible value of $d$?
1. **Modeling the Problem:**
   - Let \( m \) and \( n \) be relatively prime positive integers.
   - Consider a regular \( m \)-gon and a regular \( n \)-gon inscribed in a circle.
   - We need to find the minimum distance \( d \) in degrees between a vertex of the \( m \)-gon and a vertex of the \( n \)-gon.

2. **Angular Distance Calculation:**
   - The angular distance between consecutive vertices of the \( m \)-gon is \( \frac{360^\circ}{m} \).
   - The angular distance between consecutive vertices of the \( n \)-gon is \( \frac{360^\circ}{n} \).

3. **Minimum Distance \( d \):**
   - We need to find the minimum distance \( d \) such that \( d \) is the smallest angle between any vertex of the \( m \)-gon and any vertex of the \( n \)-gon.
   - Since \( m \) and \( n \) are relatively prime, the vertices of the \( m \)-gon and \( n \)-gon will not coincide except at the starting point (0 degrees).

4. **Using Modular Arithmetic:**
   - Consider the angular positions of the vertices of the \( m \)-gon and \( n \)-gon.
   - The vertices of the \( m \)-gon are at angles \( \frac{360k}{m} \) for \( k = 0, 1, 2, \ldots, m-1 \).
   - The vertices of the \( n \)-gon are at angles \( \frac{360j}{n} \) for \( j = 0, 1, 2, \ldots, n-1 \).

5. **Finding the Minimum Distance:**
   - We need to find the smallest positive angle \( d \) such that \( d = \left| \frac{360k}{m} - \frac{360j}{n} \right| \) for some integers \( k \) and \( j \).
   - This can be simplified to finding the smallest positive value of \( \left| \frac{360k}{m} - \frac{360j}{n} \right| \mod 360 \).

6. **Using the Property of Relatively Prime Numbers:**
   - Since \( m \) and \( n \) are relatively prime, the smallest positive value of \( \left| \frac{360k}{m} - \frac{360j}{n} \right| \) is \( \frac{360}{mn} \).
   - This is because the least common multiple (LCM) of \( m \) and \( n \) is \( mn \) when \( m \) and \( n \) are relatively prime.

7. **Conclusion:**
   - The maximum possible value of \( d \) is \( \frac{180}{mn} \).

The final answer is \( \boxed{\frac{180}{mn}} \)