A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.
1. **Identify the prime factorizations:**
   - The prime factorization of 180 is \(180 = 2^2 \cdot 3^2 \cdot 5\).
   - The prime factorization of 900 is \(900 = 2^2 \cdot 3^2 \cdot 5^2\).

2. **Analyze the conditions:**
   - Given \(\text{lcm}(x, y) = 180\), both \(x\) and \(y\) must have prime factors that do not exceed the highest powers in 180.
   - Given \(\text{lcm}(x, z) = 900\), \(z\) must include the prime factors of 900, which means \(z\) must be divisible by \(5^2 = 25\).

3. **Determine the constraints on \(z\):**
   - Since \(z\) must be divisible by 25, we can write \(z = 25k\) for some integer \(k\).
   - Both \(y\) and \(\frac{z}{5} = 5k\) must be factors of 180. Therefore, \(k\) must be a factor of \(\frac{180}{5} = 36\).

4. **List the factors of 36:**
   - The factors of 36 are \(1, 2, 3, 4, 6, 9, 12, 18, 36\).

5. **Formulate \(n\):**
   - Since \(n = \text{lcm}(y, z)\) and \(z = 25k\), we have \(n = \text{lcm}(y, 25k)\).
   - For each factor \(k\) of 36, \(z = 25k\) and \(y\) must be chosen such that \(\text{lcm}(y, 25k)\) is a valid \(n\).

6. **Verify the possible values of \(n\):**
   - For each \(k\), \(n = \text{lcm}(y, 25k)\) must be divisible by 25 and must be a multiple of \(k\).
   - Since \(k\) is a factor of 36, \(n = 25k\) is valid for each factor \(k\) of 36.

7. **Count the valid \(n\):**
   - There are 9 factors of 36, so there are 9 possible values for \(n\).

The final answer is \(\boxed{9}\)