Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$.  What is the largest integer that is a divisor of every number in $S$?
1. **Define the function and express it in a factored form:**
   \[
   f(n) = n^5 - 5n^3 + 4n = n(n-1)(n+1)(n-2)(n+2)
   \]
   This shows that \( f(n) \) is the product of five consecutive integers.

2. **Analyze divisibility by powers of 2:**
   - Among any five consecutive integers, there are at least two even numbers.
   - Among any two even numbers, one must be divisible by 4.
   - Therefore, \( f(n) \) is divisible by \( 2^3 \) (since \( 2 \times 2 \times 2 = 8 \)).
   - Example to show \( 2^4 \) does not necessarily divide \( f(n) \): For \( n = 5 \),
     \[
     f(5) = 5 \cdot 4 \cdot 6 \cdot 3 \cdot 7 = 2520
     \]
     \( 2520 \) is not divisible by \( 2^4 \).

3. **Analyze divisibility by powers of 3:**
   - Among any five consecutive integers, there are at least two numbers divisible by 3.
   - Therefore, \( f(n) \) is divisible by \( 3^2 \) (since \( 3 \times 3 = 9 \)).
   - Example to show \( 3^3 \) does not necessarily divide \( f(n) \): For \( n = 4 \),
     \[
     f(4) = 4 \cdot 3 \cdot 5 \cdot 2 \cdot 6 = 720
     \]
     \( 720 \) is not divisible by \( 3^3 \).

4. **Analyze divisibility by powers of 5:**
   - Among any five consecutive integers, there is exactly one number divisible by 5.
   - Therefore, \( f(n) \) is divisible by \( 5 \).
   - Example to show \( 5^2 \) does not necessarily divide \( f(n) \): For \( n = 5 \),
     \[
     f(5) = 5 \cdot 4 \cdot 6 \cdot 3 \cdot 7 = 2520
     \]
     \( 2520 \) is not divisible by \( 5^2 \).

5. **Analyze divisibility by primes \( p \geq 7 \):**
   - For any prime \( p \geq 7 \), there exists an \( n \) such that \( p \nmid f(n) \).
   - Example: For \( n = 3 \),
     \[
     f(3) = 3 \cdot 2 \cdot 4 \cdot 1 \cdot 5 = 120
     \]
     \( 120 \) is not divisible by primes \( p \geq 7 \).

6. **Combine the results:**
   - The largest integer that is a divisor of every number in \( S \) is:
     \[
     2^3 \cdot 3^2 \cdot 5 = 8 \cdot 9 \cdot 5 = 360
     \]

The final answer is \(\boxed{360}\)