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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 \(\boxed{500}\). 📰 Facecheck Revealed See What Your Subconscious Is Trying To Tell You 2885793 📰 Barcelona Faces Villarrello With Unexpected Lineupprivileged Edge Or Risk 4389305 📰 Grill On Fire Burn Your Teeth Like Never Beforethe Truth Revealed 6535716 📰 You Wont Believe These Hidden Resources That Save Homeless Lives Today 9788008 📰 Soobin 6883015 📰 The Ultimate Hsa Maximum Hack Double Your Healthcare Fund Without Breaking A Sweat 722879 📰 Rocket Cafe 3010758 📰 The Insane Performance Of Vfiax Vanguardheres What Happens When Ai Meets Gaming 3287054 📰 Cameron Mathison Movies And Tv Shows 1655633 📰 Ptn Stock 7159960 📰 These Top 10 Games For Online Play Will Take Your Weekend To The Next Level 6906142 📰 Shooter Cast Tv Series 3076777 📰 Muppets Haunted Mansion 399620 📰 Vallejo News 2056594 📰 Npi Login Page Not Working Instant Step By Step Solution Inside 5755158