Let \(a\) and \(b\) be positive integers satisfying \(\operatorname{lcm}(a,b)-\gcd(a,b)=23\). Find the sum of all possible values of \(a+b\).
Solution
Answer: \(129\)
Let \(d=\gcd(a,b)\), and set \(x=\frac ad\), \(y=\frac bd\). Since \(\operatorname{lcm}(a,b)=dxy\),
\[d(xy-1)=23.\]
The PDF considers two cases. If \(d=1\), then \(xy=24\), and it lists the pairs \((1,24),(2,12),(3,8),(4,6)\), or their reverses, giving sums \(25,14,11,10\). If \(d=23\), then \(xy=2\), giving \((a,b)=(46,23)\) or its reverse and sum \(69\).
The source adds these listed sums: \(10+11+14+25+69=129\).