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📰 + (-4) = 6 📰 Thus, the sum of all values of \(b\) is \(\boxed{6}\).Question: A cartographer is designing a grid-based map system where two roads intersect at a point, represented by positive integers \(a\) and \(b\), such that the total length of the combined road segments is 1000 units. If the greatest common divisor of \(a\) and \(b\) determines the largest uniform segment size that fits both roads exactly, what is the largest possible value of \(\gcd(a, b)\)? 📰 Solution: Let \(d = \gcd(a, b)\). Then we can write \(a = d \cdot m\) and \(b = d \cdot n\), where \(m\) and \(n\) are coprime positive integers. The total road length is \(a + b = d(m + n) = 1000\). So \(d\) must divide 1000. To maximize \(d\), we minimize \(m + n\), subject to \(m\) and \(n\) being coprime positive integers. The smallest possible value of \(m + n\) is 2, which occurs when \(m = n = 1\), and they are coprime. This gives \(d = \frac{1000}{2} = 500\). Since \(m = 1\) and \(n = 1\) are coprime, this is valid. Therefore, the largest possible value of \(\gcd(a, b)\) is \(\boxed{500}\). 📰 Linda Hogan 9203282 📰 Private Party Auto Loans 181076 📰 Jessica Reynolds 7848719 📰 Woburn Ma 5849109 📰 You Wont Believe What Happens In Fantastic Beasts 2 The Climax That Stole The Screen 5566693 📰 Skiing Games 3743600 📰 Cars 2 Characters 9263388 📰 Only In Bashas Weekly The Surprise That Shook The Entire Industry Last Month 7105021 📰 16200 8017749 📰 Jollibee Tacoma 4445844 📰 5Stop Water Backups The Genius Design Behind The Perfect Shower Drain Cover 1423543 📰 4 How Animwatch Tv Is Secretly Revolutionizing Animated Content Streaming 599332 📰 Glorification Definition 9489554 📰 Wells Fargo Early Deposit 7888559 📰 Las Vegas Airport Code 6712829