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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 \(\boxed{500}\). 📰 Youre Using Solventbut Did You Know Its Actually The Solute That Rules The Mix Heres Why 📰 Youve Been Missing This 📰 Youve Never Seen Sheva Alomarheres What Makes Her A Legend 📰 Youve Seen Normal Catsnow Meet The Mystery Of The Sigma Cat You Wont Believe Whats Inside 📰 Youve Seen The Smiling Through It All Memenow See Why It Spread So Fast 📰 Z 15Circ 180Circ N Quad Textor Quad Z 165Circ 180Circ N 📰 Z 15Circ 195Circ From 15Circ 180Circ N 📰 Z 165Circ 345Circ From 165Circ 180Circ N 📰 Z 210Circ Quad Z 240Circ 📰 Z 30Circ 180Circ K Quad Text Quad Z 60Circ 180Circ K 📰 Z 30Circ 60Circ 210Circ 240Circ 📰 Z 30Circ Quad Text Quad Z 60Circ 📰 Z 30Circ Quad Z 60Circ 📰 Z 4Y 1 📰 Zillions Of Style Top 5 Sideboards For Dining That Will Blend With Any Decor