Two positive integers \(x\) and \(y\) are in the ratio \(17 : 19\). If the LCM of the two numbers is \(1615\), then what is \((x + y)\) equal to?
\(180\)
Since \(x : y = 17 : 19\) and \(\gcd(17,19)=1\), let \(x = 17k\) and \(y = 19k\) for a positive integer \(k\). Then \(\text{LCM}(x,y) = 17 \times 19 \times k = 323k\). Given \(323k = 1615\), so \(k = 5\). Hence \(x = 85\), \(y = 95\), and \(x+y = 180\).
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