A circumference was divided in $n$ equal parts. On each of these parts one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$ and $17$ were in consecutive positions. In how many parts was the circumference divided?
1. **Understanding the problem**: We need to find the number of parts, \( n \), into which the circumference is divided such that the numbers \( 11 \), \( 4 \), and \( 17 \) are in consecutive positions with equal spacing.

2. **Setting up the equation**: Let \( d \) be the distance (in terms of sectors) between consecutive numbers. Since the numbers are placed in a circular manner, we can use modular arithmetic to express the positions. We know:
   \[
   4 \equiv 11 + d \pmod{n}
   \]
   \[
   17 \equiv 4 + d \pmod{n}
   \]

3. **Solving the first congruence**:
   \[
   4 \equiv 11 + d \pmod{n} \implies d \equiv 4 - 11 \pmod{n} \implies d \equiv -7 \pmod{n} \implies d \equiv n - 7 \pmod{n}
   \]

4. **Solving the second congruence**:
   \[
   17 \equiv 4 + d \pmod{n} \implies d \equiv 17 - 4 \pmod{n} \implies d \equiv 13 \pmod{n}
   \]

5. **Equating the two expressions for \( d \)**:
   \[
   n - 7 \equiv 13 \pmod{n}
   \]

6. **Simplifying the equation**:
   \[
   n - 7 = 13 + kn \quad \text{for some integer } k
   \]
   \[
   n - 7 = 13 \quad \text{(since } k = 0 \text{ is the simplest solution)}
   \]
   \[
   n = 20
   \]

7. **Verification**: We need to verify that \( n = 20 \) satisfies the conditions:
   - The distance \( d \) between consecutive numbers is \( 13 \) (from step 4).
   - Positions modulo \( 20 \):
     \[
     4 \equiv 11 + 13 \pmod{20} \implies 4 \equiv 24 \pmod{20} \implies 4 \equiv 4
     \]
     \[
     17 \equiv 4 + 13 \pmod{20} \implies 17 \equiv 17 \pmod{20}
     \]
   - Both conditions are satisfied.

The final answer is \(\boxed{20}\).