Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
1. **Define the problem and the set \( F \)**:
   - \( F \) is the set of all polynomials \( \Gamma \) such that all coefficients of \( \Gamma(x) \) are integers and \( \Gamma(x) = 1 \) has integer roots.
   - We need to find the smallest integer \( m(k) > 1 \) such that there exists \( \Gamma \in F \) for which \( \Gamma(x) = m(k) \) has exactly \( k \) distinct integer roots.

2. **Construct the polynomial \( \Gamma(x) \)**:
   - Consider the polynomial:
     \[
     \Gamma(x) = 1 + \left( \left\lfloor \frac{k}{2} \right\rfloor \right)! \left( \left\lceil \frac{k}{2} \right\rceil \right)! + (-1)^{\left\lfloor \frac{k+2}{2} \right\rfloor}(x-1)(x-2) \cdots \left(x - \left\lfloor \frac{k}{2} \right\rfloor \right) (x+1) (x+2) \cdots \left( x + \left\lceil \frac{k}{2} \right\rceil \right)
     \]

3. **Verify the roots of \( \Gamma(x) = 1 \)**:
   - The polynomial \( \Gamma(x) = 1 \) has the root \( x = 0 \).

4. **Verify the roots of \( \Gamma(x) = m(k) \)**:
   - The polynomial \( \Gamma(x) = 1 + \left( \left\lfloor \frac{k}{2} \right\rfloor \right)! \left( \left\lceil \frac{k}{2} \right\rceil \right)! \) has exactly \( k \) distinct integer roots.

5. **Determine the smallest \( m(k) \)**:
   - The smallest \( m(k) \) is given by:
     \[
     m(k) = 1 + \left( \left\lfloor \frac{k}{2} \right\rfloor \right)! \left( \left\lceil \frac{k}{2} \right\rceil \right)!
     \]

The final answer is \( \boxed{ 1 + \left( \left\lfloor \frac{k}{2} \right\rfloor \right)! \left( \left\lceil \frac{k}{2} \right\rceil \right)! } \)