Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$
1. **Understanding the Problem:**
   We need to find the maximum number of positive integers such that for any two distinct integers \(a\) and \(b\), the inequality \(|a - b| \ge \frac{ab}{100}\) holds.

2. **Rewriting the Inequality:**
   The given inequality is \(|a - b| \ge \frac{ab}{100}\). This can be rewritten in two cases:
   - \(a - b \ge \frac{ab}{100}\)
   - \(b - a \ge \frac{ab}{100}\)

3. **Analyzing the Inequality:**
   Let's consider the first case \(a - b \ge \frac{ab}{100}\):
   \[
   a - b \ge \frac{ab}{100}
   \]
   Rearrange to:
   \[
   100(a - b) \ge ab
   \]
   \[
   100a - 100b \ge ab
   \]
   \[
   100a - ab \ge 100b
   \]
   \[
   a(100 - b) \ge 100b
   \]
   \[
   a \ge \frac{100b}{100 - b}
   \]

4. **Considering the Second Case:**
   Similarly, for the second case \(b - a \ge \frac{ab}{100}\):
   \[
   b - a \ge \frac{ab}{100}
   \]
   Rearrange to:
   \[
   100(b - a) \ge ab
   \]
   \[
   100b - 100a \ge ab
   \]
   \[
   100b - ab \ge 100a
   \]
   \[
   b(100 - a) \ge 100a
   \]
   \[
   b \ge \frac{100a}{100 - a}
   \]

5. **Finding the Maximum Number of Integers:**
   To find the maximum number of integers, we need to ensure that the inequalities hold for all pairs of integers in the set. Let's test small sets of integers to see if they satisfy the condition.

6. **Testing Small Sets:**
   - For \(a = 1\) and \(b = 2\):
     \[
     |1 - 2| = 1 \quad \text{and} \quad \frac{1 \cdot 2}{100} = 0.02
     \]
     \[
     1 \ge 0.02 \quad \text{(True)}
     \]
   - For \(a = 1\), \(b = 2\), and \(c = 3\):
     \[
     |1 - 3| = 2 \quad \text{and} \quad \frac{1 \cdot 3}{100} = 0.03
     \]
     \[
     2 \ge 0.03 \quad \text{(True)}
     \]
     \[
     |2 - 3| = 1 \quad \text{and} \quad \frac{2 \cdot 3}{100} = 0.06
     \]
     \[
     1 \ge 0.06 \quad \text{(True)}
     \]

7. **Generalizing the Pattern:**
   We observe that for small integers, the condition holds. However, as the integers grow larger, the product \(ab\) increases, making \(\frac{ab}{100}\) larger. This suggests that the set of integers cannot be too large.

8. **Finding the Maximum Set:**
   By testing further, we find that the maximum set of integers that satisfy the condition is \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). For any larger set, the condition \(|a - b| \ge \frac{ab}{100}\) will not hold for some pairs.

The final answer is \(\boxed{10}\).