14.A. Choose $n$ numbers from $1,2, \cdots, 9$. Among them, there must be some numbers (at least one, or possibly all) whose sum is divisible by 10. Find the minimum value of $n$.
14. A. When $n=4$, the numbers $1,3,5,8$ do not have any subset of numbers whose sum is divisible by 10.

When $n=5$, let $a_{1}, a_{2}, \cdots, a_{5}$ be five different numbers from $1,2, \cdots, 9$. If the sum of any subset of these numbers cannot be divisible by 10, then $a_{1}, a_{2}, \cdots, a_{5}$ cannot simultaneously contain 1 and 9, 2 and 8, 3 and 7, 4 and 6. Therefore, $a_{1}, a_{2}, \cdots, a_{5}$ must contain the number 5.

If $a_{1}, a_{2}, \cdots, a_{5}$ contains 1, then it does not contain 9. Thus, it does not contain $4 (4+1+5=10)$, so it contains 6; thus, it does not contain $3 (3+6+1=10)$, so it contains 7; thus, it does not contain $2 (2+1+7=10)$, so it contains 8. However, $5+7+8=20$ is a multiple of 10, which is a contradiction.

If $a_{1}, a_{2}, \cdots, a_{5}$ contains 9, then it does not contain 1. Thus, it does not contain $6 (6+9+5=20)$, so it contains 4; thus, it does not contain $7 (7+4+9=20)$, so it contains 3; thus, it does not contain $8 (8+9+3=10)$, so it contains 2. However, $5+3+2=10$ is a multiple of 10, which is a contradiction.
In summary, the minimum value of $n$ is 5.