Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$.

$\textit{Proposed by Pablo Serrano, Ecuador}$
1. **Identify the problem constraints and initial conditions:**
   - Lucía multiplies some positive one-digit numbers and obtains a number \( n \) greater than 10.
   - She then multiplies all the digits of \( n \) and obtains an odd number.
   - We need to find all possible values of the units digit of \( n \).

2. **Analyze the units digit of \( n \):**
   - The units digit of \( n \) must be such that when all digits of \( n \) are multiplied, the result is an odd number.
   - For the product of the digits of \( n \) to be odd, all digits of \( n \) must be odd (since the presence of any even digit would make the product even).

3. **Possible odd digits:**
   - The odd digits are \( 1, 3, 5, 7, \) and \( 9 \).

4. **Consider the units digit of \( n \):**
   - If the units digit of \( n \) is \( 1, 3, 7, \) or \( 9 \), then \( n \) must be of the form \( 3^a \cdot 7^b \) (since any factors of \( 2 \) or \( 5 \) would make the units digit \( 0, 2, 4, 6, 8 \) or \( 0, 5 \), respectively).

5. **Analyze the form \( 3^a \cdot 7^b \):**
   - We need to check the possible units digits of \( 3^a \cdot 7^b \) modulo 20.
   - Notice that \( 3^a \cdot 7^b \equiv 21^b \cdot 3^{a-b} \equiv 3^{a-b} \pmod{20} \).

6. **Calculate the units digits of powers of 3 modulo 20:**
   - \( 3^0 \equiv 1 \pmod{20} \)
   - \( 3^1 \equiv 3 \pmod{20} \)
   - \( 3^2 \equiv 9 \pmod{20} \)
   - \( 3^3 \equiv 27 \equiv 7 \pmod{20} \)
   - \( 3^4 \equiv 81 \equiv 1 \pmod{20} \)

7. **Possible units digits of \( n \):**
   - From the above, \( n \equiv 1, 3, 9, 7 \pmod{20} \).

8. **Check the tens digit condition:**
   - If the tens digit of \( n \) is odd, then the remainder when \( n \) is divided by 20 would have an odd tens digit, which is a contradiction since the product of the digits of \( n \) must be odd.
   - Therefore, \( n \) must have an even tens digit.

9. **Conclusion:**
   - The units digit of \( n \) cannot be \( 1, 3, 7, \) or \( 9 \) because the tens digit must be even.
   - The only remaining possibility is that the units digit of \( n \) is \( 5 \), which works for \( n = 3 \cdot 5 = 15 \).

The final answer is \(\boxed{5}\)