Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$.
1. We need to find the smallest positive integer \( n \) such that there exists an integer \( m \) for which \(\left\lfloor \frac{10^n}{m} \right\rfloor = 1989\).
2. This implies the inequality:
   \[
   1989 \leq \frac{10^n}{m} < 1990
   \]
   which can be rewritten as:
   \[
   \frac{10^n}{1990} < m \leq \frac{10^n}{1989}
   \]
3. We will check for the smallest \( n \) starting from \( n = 1 \) and increasing until we find a suitable \( m \).

4. For \( n = 1 \):
   \[
   \frac{10^1}{1990} < m \leq \frac{10^1}{1989}
   \]
   \[
   \frac{10}{1990} < m \leq \frac{10}{1989}
   \]
   \[
   0.005 < m \leq 0.00503
   \]
   No integer \( m \) satisfies this inequality.

5. For \( n = 2 \):
   \[
   \frac{10^2}{1990} < m \leq \frac{10^2}{1989}
   \]
   \[
   \frac{100}{1990} < m \leq \frac{100}{1989}
   \]
   \[
   0.05025 < m \leq 0.05028
   \]
   No integer \( m \) satisfies this inequality.

6. For \( n = 3 \):
   \[
   \frac{10^3}{1990} < m \leq \frac{10^3}{1989}
   \]
   \[
   \frac{1000}{1990} < m \leq \frac{1000}{1989}
   \]
   \[
   0.5025 < m \leq 0.5028
   \]
   No integer \( m \) satisfies this inequality.

7. For \( n = 4 \):
   \[
   \frac{10^4}{1990} < m \leq \frac{10^4}{1989}
   \]
   \[
   \frac{10000}{1990} < m \leq \frac{10000}{1989}
   \]
   \[
   5.025 < m \leq 5.028
   \]
   No integer \( m \) satisfies this inequality.

8. For \( n = 5 \):
   \[
   \frac{10^5}{1990} < m \leq \frac{10^5}{1989}
   \]
   \[
   \frac{100000}{1990} < m \leq \frac{100000}{1989}
   \]
   \[
   50.25 < m \leq 50.28
   \]
   No integer \( m \) satisfies this inequality.

9. For \( n = 6 \):
   \[
   \frac{10^6}{1990} < m \leq \frac{10^6}{1989}
   \]
   \[
   \frac{1000000}{1990} < m \leq \frac{1000000}{1989}
   \]
   \[
   502.51 < m \leq 502.76
   \]
   No integer \( m \) satisfies this inequality.

10. For \( n = 7 \):
    \[
    \frac{10^7}{1990} < m \leq \frac{10^7}{1989}
    \]
    \[
    \frac{10000000}{1990} < m \leq \frac{10000000}{1989}
    \]
    \[
    5025.1256 < m \leq 5027.1386
    \]
    Here, \( m \) can be 5026 or 5027.

11. Verify:
    - For \( m = 5026 \):
      \[
      \left\lfloor \frac{10^7}{5026} \right\rfloor = \left\lfloor 1989.004 \right\rfloor = 1989
      \]
    - For \( m = 5027 \):
      \[
      \left\lfloor \frac{10^7}{5027} \right\rfloor = \left\lfloor 1989.002 \right\rfloor = 1989
      \]

Thus, the smallest \( n \) for which there exists an integer \( m \) such that \(\left\lfloor \frac{10^n}{m} \right\rfloor = 1989\) is \( n = 7 \).

The final answer is \( \boxed{ n = 7 } \)