Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k,$ the integer 
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016.$

1. **Understanding the Problem:**
   We need to find the smallest positive integer \( j \) such that for every polynomial \( p(x) \) with integer coefficients and for every integer \( k \), the \( j \)-th derivative of \( p(x) \) at \( k \) is divisible by \( 2016 \).

2. **Expression for the \( j \)-th Derivative:**
   Consider a polynomial \( p(x) = \sum_{i=0}^n a_i x^i \). The \( j \)-th derivative of \( p(x) \) is given by:
   \[
   p^{(j)}(x) = \sum_{i=j}^n a_i \frac{i!}{(i-j)!} x^{i-j}
   \]
   Evaluating this at \( x = k \), we get:
   \[
   p^{(j)}(k) = \sum_{i=j}^n a_i \frac{i!}{(i-j)!} k^{i-j}
   \]

3. **Special Case for \( p(x) = x^j \):**
   Consider the polynomial \( p(x) = x^j \). The \( j \)-th derivative of \( p(x) \) is:
   \[
   p^{(j)}(x) = j!
   \]
   Evaluating this at \( x = k \), we get:
   \[
   p^{(j)}(k) = j!
   \]
   For \( p^{(j)}(k) \) to be divisible by \( 2016 \), we need \( j! \) to be divisible by \( 2016 \).

4. **Prime Factorization of 2016:**
   The prime factorization of \( 2016 \) is:
   \[
   2016 = 2^5 \cdot 3^2 \cdot 7
   \]
   Therefore, \( j! \) must include the factors \( 2^5 \), \( 3^2 \), and \( 7 \).

5. **Finding the Smallest \( j \):**
   We need to find the smallest \( j \) such that \( j! \) includes all the prime factors of \( 2016 \):
   - \( 2^5 \) requires at least \( j \geq 5 \)
   - \( 3^2 \) requires at least \( j \geq 6 \)
   - \( 7 \) requires at least \( j \geq 7 \)

   Therefore, the smallest \( j \) that satisfies all these conditions is \( j = 7 \).

6. **Verification:**
   - For \( j = 7 \), \( 7! = 5040 \)
   - \( 5040 \) is divisible by \( 2016 \) since:
     \[
     5040 = 2^4 \cdot 3^2 \cdot 5 \cdot 7
     \]
     This includes all the prime factors of \( 2016 \).

7. **Conclusion:**
   The smallest positive integer \( j \) such that for every polynomial \( p(x) \) with integer coefficients and for every integer \( k \), the \( j \)-th derivative of \( p(x) \) at \( k \) is divisible by \( 2016 \) is \( j = 8 \).

The final answer is \( \boxed{8} \).