Find all positive integers $n$ so that both $n$ and $n + 100$ have odd numbers of divisors.
To find all positive integers \( n \) such that both \( n \) and \( n + 100 \) have an odd number of divisors, we need to understand the conditions under which a number has an odd number of divisors. A number has an odd number of divisors if and only if it is a perfect square. This is because the divisors of a number usually come in pairs, except when the number is a perfect square (where one of the divisors is repeated).

1. Let \( n = a^2 \) and \( n + 100 = b^2 \) for some integers \( a \) and \( b \). This implies:
   \[
   b^2 - a^2 = 100
   \]
   Using the difference of squares, we can rewrite this as:
   \[
   (b - a)(b + a) = 100
   \]

2. We need to find pairs \((b - a)\) and \((b + a)\) that multiply to 100 and have the same parity (both even or both odd) to ensure \( b \) and \( a \) are integers.

3. The factor pairs of 100 are:
   \[
   (1, 100), (2, 50), (4, 25), (5, 20), (10, 10)
   \]
   Among these, the pairs that have the same parity are \((2, 50)\) and \((10, 10)\).

4. For the pair \((2, 50)\):
   \[
   b - a = 2 \quad \text{and} \quad b + a = 50
   \]
   Solving these equations simultaneously:
   \[
   b - a = 2 \quad \text{(1)}
   \]
   \[
   b + a = 50 \quad \text{(2)}
   \]
   Adding equations (1) and (2):
   \[
   2b = 52 \implies b = 26
   \]
   Subtracting equation (1) from (2):
   \[
   2a = 48 \implies a = 24
   \]
   Therefore, \( n = a^2 = 24^2 = 576 \).

5. For the pair \((10, 10)\):
   \[
   b - a = 10 \quad \text{and} \quad b + a = 10
   \]
   Solving these equations simultaneously:
   \[
   b - a = 10 \quad \text{(3)}
   \]
   \[
   b + a = 10 \quad \text{(4)}
   \]
   Adding equations (3) and (4):
   \[
   2b = 20 \implies b = 10
   \]
   Subtracting equation (3) from (4):
   \[
   2a = 0 \implies a = 0
   \]
   Therefore, \( n = a^2 = 0^2 = 0 \), but \( n \) must be a positive integer, so this solution is invalid.

Conclusion:
The only valid solution is \( n = 576 \).

The final answer is \( \boxed{576} \).