Question: A cartographer uses a coordinate system where two perpendicular roads are aligned with integer-length segments \(a\) and \(b\) such that \(a + b = 2025\) and \(\gcd(a, b)\) is as large as possible. What is the largest possible value of \(\gcd(a, b)\)?

["Title: Maximize the GCD of Two Perpendicular Road Segments: How to Choose (a) and (b) When (a + b = 2025)", "Meta Description:\nDiscover the maximum possible value of (\gcd(a, b)) when two perpendicular roads measure integer-length segments (a) and (b) such that (a + b = 2025). Learn how aligning (a) and (b) with a common divisor optimizes their GCD.", "---", "Introduction\nWhen designing grid-like road systems—especially where perpendicular segments form the foundation of urban planning—cartographers often seek optimal configurations. Given a fixed total length of (a + b = 2025), involving two positive integers (a) and (b) such that their greatest common divisor (\gcd(a, b)) is maximized, what is the largest possible GCD?", "This article explores how to maximize (\gcd(a, b)) under the constraint (a + b = 2025), revealing strategic number theory insights every mapmaker and programmer should know.", "---", "Understanding the Problem\nLet (a) and (b) be positive integers satisfying:\n[\na + b = 2025\n]\nand we aim to maximize:\n[\n\gcd(a, b)\n]", "Let (d = \gcd(a, b)). Since (d) divides both (a) and (b), we can write:\n[\na = d \cdot m,\quad b = d \cdot n\n]\nwhere (m) and (n) are coprime integers ((\gcd(m, n) = 1)). Substituting into the sum:\n[\nd(m + n) = 2025\n]\nThus, (d) must be a divisor of 2025, and (m + n = \frac{2025}{d}).", "To maximize (d), we need the largest divisor (d) such that (\frac{2025}{d}) can be expressed as the sum of two coprime positive integers (m) and (n).", "---", "Key Insight: When Can (m + n = s) Be Expressed with Coprime Integers?\nFor any integer (s \geq 2), it’s always possible to find coprime positive integers (m) and (n) such that (m + n = s). For example:\n- (m = 1), (n = s - 1); but (\gcd(1, s-1) = 1), so coprime.\n- More generally, (m = \lfloor s/2 \rfloor), (n = s - \lfloor s/2 \rfloor) often yields coprime pairs, especially when (s) is not a power of 2 (but even then, coprime pairs exist).", "Thus, any positive integer (s \geq 2) allows a coprime decomposition (m + n = s).", "Therefore, the only restriction on (d) is that (d) divides 2025 and (\frac{2025}{d} \geq 2) (since (m, n \geq 1)).", "---", "Step 1: Factorize 2025\nTo find all divisors, factorize:\n[\n2025 = 25 \ imes 81 = 5^2 \ imes 3^4\n]\nSo the number of positive divisors is ((2+1)(4+1) = 15). The full list of divisors is generated from:\n[\n3^a \cdot 5^b,\quad \ ext{where } 0 \leq a \leq 4,; 0 \leq b \leq 2\n]\nWe seek the largest divisor (d) such that (\frac{2025}{d} = m + n \geq 2). Since all divisors satisfy this (as (m+n = 2025/d \geq 2) for (d \leq 1012)), the maximum possible (d) is the largest divisor of 2025 less than or equal to (2025/2 = 1012.5).", "So we need the largest proper divisor of 2025 less than or equal to 1012.", "---", "Step 2: Find Largest Divisor ≤ 1012\nThe full divisors near 1012:", "- Try (d = 675 = 3^3 \cdot 5^2 = 27 \cdot 25 = 675)\n- Check: (2025 / 675 = 3), so (m + n = 3), possible coprime pairs: (1,2) or (2,1), (\gcd = 1). So valid.\n- Is there a larger divisor?\n - Next divisor up from 675: try (3^4 \cdot 5 = 81 \cdot 5 = 405) → smaller\n - (3^4 \cdot 5^2 = 81 \cdot 25 = 2025) → too big ((d = 2025) → (s = 1), invalid)\n - (3^2 \cdot 5^2 = 9 \cdot 25 = 225) → smaller\n - (3^4 = 81), etc. — all less than 675 except possibly ( \gcd = 2025 ), but (s = 1) impossible", "But wait: (d = 675) gives (s = 3), which is acceptable.\nIs there a divisor between 675 and 1012?", "List higher candidates:\n- (3^4 \cdot 5 = 81 \cdot 5 = 405)\n- (3^3 \cdot 5^2 = 27 \cdot 25 = 675)\n- (3^2 \cdot 5^2 = 9 \cdot 25 = 225)\n- (3^4 \cdot 5 = 405)\n- (3^1 \cdot 5^3) — invalid, not a divisor\n- (3^0 \cdot 5^3) — invalid", "Next larger than 675?\nTry (3^4 \cdot 5 = 405), (3^3 \cdot 5^2 = 675), (3^2 \cdot 5^2 = 225) — no divisor between 675 and 1012.", "But what about (d = 405)? Smaller than 675 — not better.", "Wait — try (d = 2025 / s), where (s = m + n \geq 2), so smallest (s = 2) → largest (d = 2025 / 2 = 1012.5) → not integer\nNext possible: (s = 3) → (d = 675) — largest possible valid divisor.", "Hence, maximum possible (d = 675).", "---", "Step 3: Validate That (\gcd(a, b) = 675) Is Achievable\nLet (d = 675), so (a = 675m), (b = 675n), with (m + n = 3), (\gcd(m,n) = 1).", "Possible pair: (m = 1), (n = 2) (or vice versa). Then:\n[\na = 675 \cdot 1 = 675,\quad b = 675 \cdot 2 = 1350\n]\nCheck: (a + b = 675 + 1350 = 2025), good.\n(\gcd(675, 1350)):\n[\n\gcd(675, 1350) = 675 \cdot \gcd(1, 2) = 675 \cdot 1 = 675\n]\nValid.", "Can we get higher? Try (d = 1012.5) — not integer. Next integer divisor below 1012 is 675. So no.", "What about (d = 405)? Smaller.", "Thus, 675 is the maximum possible GCD.", "---", "Why This Matters in Cartography\nAligning road segments with maximal common divisors optimizes symmetry, modular planning, and construction standardization. For fixed total length, maximizing (\gcd(a,b)) ensures balanced, repeatable design—ideal for city grids or military layouts where efficiency and regularity are key.", "---", "Conclusion\nTo maximize (\gcd(a, b)) with (a + b = 2025) and (a, b) positive integers:\n- Factor 2025: (3^4 \cdot 5^2)\n- The maximum divisor (d) such that (2025/d \geq 2) is (d = 675)\n- Achievable via (a = 675), (b = 1350) with (\gcd = 675)", "Thus, the largest possible value of (\gcd(a, b)) is:\n[\n\boxed{675}\n]", "---", "FAQ\nQ: Can any (a + b) value achieve the maximum (\gcd)?\nA: Only if (a + b) is divisible by a number (d) such that (s = (a+b)/d \geq 2) and (s) permits coprime (m, n). Since 2025 is divisible by 675 and (2025/675 = 3 \geq 2), it works.", "Q: Is 675 the only maximizing pair?\nA: No, other divisors like 225 also work but yield smaller GCDs. Only (d = 675) gives (\gcd = 675).", "Q: What if (a) and (b) are equal?\nA: Then (a = b = 1012.5), not integers. So equality is impossible. Max GCD occurs when they’re unequal but"]