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Let n be a positive integer such that n= ABC_7 and n=CBA_11. Find the largest possible value of n in base 10.

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To solve this problem, we will interpret the given representations of the number \( n \) in different bases. Let us denote \( A \), \( B \), and \( C \) as the digits in their respective base representations.

  1. Understanding \( n \) in base 7: The representation \( n = ABC_7 \) means: \[ n = A \cdot 7^2 + B \cdot 7^1 + C \cdot 7^0 = 49A + 7B + C \]

  2. Understanding \( n \) in base 11: The representation \( n = CBA_{11} \) means: \[ n = C \cdot 11^2 + B \cdot 11^1 + A \cdot 11^0 = 121C + 11B + A \]

  3. Setting the two equations equal: Since both expressions represent \( n \), we get the equation: \[ 49A + 7B + C = 121C + 11B + A \] Rearranging this gives: \[ 49A - A + 7B - 11B + C - 121C = 0 \] Simplifying further leads to: \[ 48A - 4B - 120C = 0 \] Dividing through by 4, we obtain: \[ 12A - B - 30C = 0 \quad \text{or} \quad B = 12A - 30C \]

  4. Constraints on \( A \), \( B \), and \( C \):

    • Since \( ABC_7 \) indicates \( A \), \( B \), and \( C \) are digits in base 7, we have \( 0 \leq A, B, C < 7 \).
    • Since \( CBA_{11} \) indicates \( C \), \( B \), and \( A \) can be digits in base 11, we thus have \( 0 \leq C, B, A < 11 \).
    • Since \( n \) is in base 7, \( A \) cannot be 0 (it must be a positive integer), so \( 1 \leq A < 7 \).

  5. Evaluating possible values: We now search for integer values of \( A \) and \( C \):

    • If \( C = 0 \): \[ B = 12A \quad \text{(not valid since } B < 7\text{)} \]
    • If \( C = 1 \): \[ B = 12A - 30 \quad \text{(must have } 0 \leq B < 7\text{). We simplify:} \] \[ 30 \leq 12A < 37 \Rightarrow 2.5 \leq A < 3.083, \text{ thus } A = 3 \Rightarrow B = 6. \]
    • If \( C = 2 \): \[ B = 12A - 60 \quad \text{(must have } 0 \leq B < 7\text{). We simplify:} \] \[ 60 \leq 12A < 67 \Rightarrow 5 \leq A < 5.583 \Rightarrow A = 5 \Rightarrow B = 0. \]
    • If \( C = 3 \): \[ B = 12A - 90 \quad \Rightarrow 90 < 12A < 97 \Rightarrow A < 8.083 \text{ (not valid for A in base 7)}. \]

  6. Valid Combinations: The only valid combination is \( A = 5 \), \( B = 0 \), and \( C = 2 \). We can now compute \( n \): \[ n = ABC_7 = 5 \cdot 49 + 0 \cdot 7 + 2 = 245 + 0 + 2 = 247. \]

  7. Rechecking base 11: Now check it in base 11: \[ n = CBA_{11} = 2 \cdot 121 + 0 \cdot 11 + 5 = 242 + 0 + 5 = 247. \]

  8. Conclusion: The largest possible value of \( n \) is

\[ \boxed{247}. \]