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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 \(\boxed{500}\). 📰 Transform Your Slides Instantlyheres How To Add Eye Catching Fonts To Powerpoint Fast 2742539 📰 Unlock Massive Returns How Fidelity Investments In Monterey Ca Is Changing The Game 8710064 📰 Hyatt Place Lax 8464633 📰 How An Orchid Mantis Tricks Hunters Without A Single Needle 2506128 📰 From Fans To Virality The Complete Story Behind Lady Devil May Crys Explosive Rise 3224649 📰 The Milking Maid Look Thats Sweeping Fashion Blogsget Yours Before Its Gone 6527824 📰 Print Powerpoint With Notes This One Tool Changes Everything Forever 4476193 📰 Copper Hair This Rare Look Is Taking The Beauty World By Storm 1494061 📰 Crimson Court 1444268 📰 Master Online Gameplay Without Downloadsbest Browser Multiplayer Games Revealed 31341 📰 Zoophilia Pokemon 6870652 📰 Create A Windows Install Usb 1577827 📰 Billie Eilish Converse 3351728 📰 Wwi Trench Game Beats The Battlefield With Strategic Survival Mechanics 1247280 📰 Women Of Brewster Place Cast 7450585