If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, what is the largest possible value of $n$? 

$ 
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{None of the preceding}
$
To solve this problem, we need to determine the largest integer \( n \) such that the integers \( 1, 2, \ldots, n \) can be divided into two sets where neither set contains the arithmetic mean of any two of its elements.

1. **Understanding the Problem:**
   - We need to partition the set \( \{1, 2, \ldots, n\} \) into two subsets \( A \) and \( B \) such that for any \( a, b \in A \), the arithmetic mean \( \frac{a+b}{2} \notin A \), and similarly for \( B \).

2. **Testing Small Values:**
   - For \( n = 1 \): Trivially, the set \( \{1\} \) can be partitioned into \( A = \{1\} \) and \( B = \emptyset \).
   - For \( n = 2 \): The set \( \{1, 2\} \) can be partitioned into \( A = \{1\} \) and \( B = \{2\} \).
   - For \( n = 3 \): The set \( \{1, 2, 3\} \) can be partitioned into \( A = \{1, 3\} \) and \( B = \{2\} \).

3. **Generalizing the Pattern:**
   - We need to check if this pattern holds for larger values of \( n \). Let's consider \( n = 9 \) and \( n = 10 \).

4. **Checking \( n = 9 \):**
   - Assume we can partition \( \{1, 2, \ldots, 9\} \) into two sets \( A \) and \( B \).
   - Without loss of generality, let \( 5 \in A \). Then, one of \( 4 \) or \( 6 \) must also be in \( A \). Assume \( 4 \in A \).
   - Since \( 4, 5 \in A \), \( 3 \) and \( 6 \) must be in \( B \).
   - Since \( 6 \in B \), \( 9 \in A \) (because \( 6, 9 \in B \) would imply \( 7.5 \in B \), which is not an integer).
   - Since \( 5, 9 \in A \), \( 7 \in B \).
   - Since \( 6, 7 \in B \), \( 8 \in A \).
   - Since \( 5, 8 \in A \), \( 2 \in B \).
   - Since \( 5, 9 \in A \), \( 1 \in B \).
   - This results in \( B = \{1, 2, 3, 6, 7\} \), which violates the condition because \( 1, 3 \in B \) implies \( 2 \in B \).

5. **Checking \( n = 8 \):**
   - Assume we can partition \( \{1, 2, \ldots, 8\} \) into two sets \( A \) and \( B \).
   - Let \( 5 \in A \). Then, one of \( 4 \) or \( 6 \) must also be in \( A \). Assume \( 4 \in A \).
   - Since \( 4, 5 \in A \), \( 3 \) and \( 6 \) must be in \( B \).
   - Since \( 6 \in B \), \( 8 \in A \).
   - Since \( 5, 8 \in A \), \( 2 \in B \).
   - Since \( 4, 6 \in B \), \( 7 \in A \).
   - Since \( 5, 7 \in A \), \( 1 \in B \).
   - This results in \( A = \{4, 5, 7, 8\} \) and \( B = \{1, 2, 3, 6\} \), which satisfies the condition.

Therefore, the largest possible value of \( n \) is \( 8 \).

The final answer is \( \boxed{8} \)