The divisors of 2023 are \(1, 7, 17, 49, 119, 289, 2023\). The largest divisor is 2023 itself. If \(d = 2023\), then \(x + y = 1\), which implies one of \(x\) or \(y\) is 0, contradicting \(x, y\) being positive integers. So, the largest feasible \(d\) is 289.

["Understanding the Divisors of 2023: A Mathematical Exploration", "The number 2023 has a unique set of divisors: (1, 7, 17, 49, 119, 289, 2023). Among these, each divides 2023 exactly and plays a key role in number theory and factorization. In this article, we explore the divisors of 2023 in detail, identify why 2023 cannot be the largest feasible divisor in many practical applications, and explain how proper constraints lead us to the most meaningful divisor—289.", "---", "### The Complete Set of Divisors", "The divisors of 2023 are:\n- (1)\n- (7)\n- (17)\n- (49)\n- (119)\n- (289)\n- (2023)", "These numbers reflect the complete factorization of 2023, which is (7 \ imes 17^2).", "---", "### Why 2023 Itself Is Not a Useful Divisor in Many Contexts", "Mathematically, the divisors of a number include 1 and itself: for any integer (n), (1) and (n) always divide (n). However, in practical scenarios—especially when selecting divisors for pairing into factor pairs like (d) and (x+y = \frac{n}{d})—using 2023 leads to an impractical result.", "Suppose (d = 2023). Then according to the equation\n[\nx + y = \frac{2023}{2023} = 1\n]\nSince (x) and (y) are positive integers, (x + y = 1) implies one of them must be 0—which violates the requirement that both are positive. Thus, 2023 cannot act as a valid divisor in contexts requiring positive integer solutions for (x, y).", "---", "### The Largest Practical Divisor: 289", "Given this limitation, the feasible largest divisor to consider is 289, which is (17^2). Let’s verify this choice:", "If (d = 289), then\n[\nx + y = \frac{2023}{289} = 7\n]\nSince 7 is a positive integer, we can choose pairs like (x = 1, y = 6) (or vice versa), satisfying all constraints.", "Thus, 289 is the largest divisor of 2023 for which (x + y) is positive and integral, making it the optimal choice.", "---", "### The Full List of Valid Divisors (Excluding Invalid Case)", "| Divisor | Validity of (x + y = \frac{2023}{d}) |\n|---------|------------------------------------------|\n| (1) | Invalid ((x + y = 2023), but (x, y > 0) impossible) |\n| (7) | Valid ((x + y = 289)) |\n| (17) | Valid ((x + y = 119)) |\n| (49) | Valid ((x + y = 41.28) → Wait, error? Let’s check.) |", "Wait: (2023 \div 49 = 41.2857), which is not an integer! This contradicts our earlier list — clearly, 49 does not divide 2023 exactly. Let’s correct and clarify.", "Hold on — there is a mistake in the earlier claimed divisors.", "---", "### Correct Factor Analysis of 2023", "Let’s verify the actual divisors of 2023:\nFactorize (2023 = 7 \ imes 17 \ imes 17 = 7 \ imes 17^2).", "Thus, the positive divisors are formed by (7^a \cdot 17^b) where (a = 0) or (1), (b = 0, 1, 2):", "- (7^0 \cdot 17^0 = 1)\n- (7^1 \cdot 17^0 = 7)\n- (7^0 \cdot 17^1 = 17)\n- (7^1 \cdot 17^1 = 119)\n- (7^0 \cdot 17^2 = 289)\n- (7^1 \cdot 17^2 = 2023)", "So the correct set of divisors is:\n[\n1, 7, 17, 119, 289, 2023\n]", "Note: (49 = 7^2) does not divide 2023, since (7^2 = 49) does not appear in the prime factorization.", "---", "### Final Summary: Valid Lastest Divisors", "Thus, valid divisors of 2023 are:\n[\n1, 7, 17, 119, 289, 2023\n]", "The largest divisor (d) for which (x + y = \frac{2023}{d}) yields positive integers is indeed 289:", "[\nx + y = \frac{2023}{289} = 7 \quad \ ext{(valid)}\n]", "For (d = 2023), (x + y = 1) → requires one zero → invalid.", "---", "### Conclusion", "Understanding which divisors of a number are truly valid in context—especially when constraint-based pairing applies—is essential. While 2023 contains many divisors, 289 is the largest feasible one that supports positive integer pairs ((x, y)).", "This exploration not only clarifies the divisor structure but also underscores how mathematical rigor refines practical number use—turning raw divisors into meaningful, usable solutions.", "---", "Keywords: divisors of 2023, factorization of 2023, mathematical divisors, positive integer pairs, 2023 number theory, divisor pairing, largest feasible divisor, 289 divisor, SOLUTION: The correct divisors are (1, 7, 17, 119, 289, 2023), with 289 as the largest usable divisor."]