Find all positive integers $n$ such that we can divide the set $\{1,2,3,\ldots,n\}$ into three sets with the same sum of members.
To find all positive integers \( n \) such that the set \(\{1, 2, 3, \ldots, n\}\) can be divided into three subsets with the same sum of members, we need to follow these steps:

1. **Sum of the first \( n \) positive integers:**
   The sum of the first \( n \) positive integers is given by:
   \[
   S = \frac{n(n+1)}{2}
   \]
   For this sum to be divisible by 3 (since we want to divide it into three subsets with equal sum), we must have:
   \[
   3 \mid \frac{n(n+1)}{2}
   \]

2. **Divisibility condition:**
   For \( \frac{n(n+1)}{2} \) to be divisible by 3, either \( n \) or \( n+1 \) must be divisible by 3. This implies:
   \[
   n \equiv 0 \pmod{3} \quad \text{or} \quad n \equiv 2 \pmod{3}
   \]

3. **Minimum value of \( n \):**
   Additionally, we need to ensure that \( n \) is large enough to allow for the division into three subsets. The smallest \( n \) that satisfies the condition \( 3 \mid \frac{n(n+1)}{2} \) and allows for non-trivial subsets is \( n \geq 5 \).

4. **Constructing subsets for specific values of \( n \):**
   - For \( n = 5 \):
     \[
     A_5 = \{1, 4\}, \quad B_5 = \{2, 3\}, \quad C_5 = \{5\}
     \]
     Sum of each subset: \( 5 \)

   - For \( n = 6 \):
     \[
     A_6 = \{1, 6\}, \quad B_6 = \{2, 5\}, \quad C_6 = \{3, 4\}
     \]
     Sum of each subset: \( 7 \)

   - For \( n = 8 \):
     \[
     A_8 = \{1, 2, 3, 6\}, \quad B_8 = \{4, 8\}, \quad C_8 = \{5, 7\}
     \]
     Sum of each subset: \( 12 \)

   - For \( n = 9 \):
     \[
     A_9 = \{1, 2, 3, 4, 5\}, \quad B_9 = \{6, 9\}, \quad C_9 = \{7, 8\}
     \]
     Sum of each subset: \( 15 \)

5. **General construction for \( n+6 \):**
   If we have subsets \( A_n, B_n, C_n \) for some \( n \), we can construct subsets for \( n+6 \) as follows:
   \[
   A_{n+6} = A_n \cup \{n+1, n+6\}, \quad B_{n+6} = B_n \cup \{n+2, n+5\}, \quad C_{n+6} = C_n \cup \{n+3, n+4\}
   \]
   This construction ensures that the sum of each subset remains equal.

Conclusion:
The positive integers \( n \) such that the set \(\{1, 2, 3, \ldots, n\}\) can be divided into three subsets with the same sum of members are those that satisfy:
\[
n \equiv 0 \pmod{3} \quad \text{or} \quad n \equiv 2 \pmod{3} \quad \text{and} \quad n \geq 5
\]

The final answer is \( \boxed{ n \equiv 0 \pmod{3} } \) or \( n \equiv 2 \pmod{3} \) and \( n \geq 5 \).