An integer $n\ge1$ is [i]good [/i] if the following property is satisfied:
If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$.
How many good integers are $n\ge 1$?
1. **Understanding the Problem:**
   We need to find the number of integers \( n \ge 1 \) such that if a positive integer is divisible by each of the nine numbers \( n+1, n+2, \ldots, n+9 \), then it is also divisible by \( n+10 \).

2. **Rephrasing the Condition:**
   The condition can be rephrased using the least common multiple (LCM). Specifically, we need:
   \[
   \text{If } k \text{ is divisible by } \text{lcm}(n+1, n+2, \ldots, n+9), \text{ then } k \text{ is also divisible by } n+10.
   \]
   This implies:
   \[
   n+10 \mid \text{lcm}(n+1, n+2, \ldots, n+9).
   \]

3. **Finding the LCM:**
   We need to find the LCM of the numbers \( n+1, n+2, \ldots, n+9 \). The LCM of these numbers must be divisible by \( n+10 \).

4. **Considering the Range of \( n \):**
   Since \( n \ge 1 \), the smallest value of \( n+10 \) is 11. We need to check for which values of \( n \), \( n+10 \) divides the LCM of \( n+1, n+2, \ldots, n+9 \).

5. **Checking Divisibility:**
   We need to check the divisibility condition for each \( n \) such that \( n+10 \) divides the LCM of \( n+1, n+2, \ldots, n+9 \). This can be done by iterating through possible values of \( n \) and checking the condition.

6. **Simplifying the Problem:**
   We can simplify the problem by noting that the LCM of any set of 9 consecutive integers is always divisible by the largest integer in the set. Therefore, we need to check if \( n+10 \) is one of the numbers in the set \( \{n+1, n+2, \ldots, n+9\} \).

7. **Finding the Valid \( n \):**
   For \( n+10 \) to be one of \( n+1, n+2, \ldots, n+9 \), we need:
   \[
   n+10 \in \{n+1, n+2, \ldots, n+9\}.
   \]
   This is impossible because \( n+10 \) is always greater than \( n+9 \).

8. **Conclusion:**
   Since \( n+10 \) cannot be one of the numbers \( n+1, n+2, \ldots, n+9 \), there are no integers \( n \ge 1 \) that satisfy the given condition.

The final answer is \( \boxed{0} \).