Problem
NT-B1-M02-P022 Pairs with Given GCD and LCM
#22
★★★★☆ Level 4 of 5
Find all unordered pairs of positive integers \((a,b)\) such that \(\gcd(a,b)=12\) and \(\operatorname{lcm}(a,b)=720\).
Set \(a=12x\), \(b=12y\), where \(\gcd(x,y)=1\).
Let \(a=12x\), \(b=12y\), \(\gcd(x,y)=1\). Then \(\operatorname{lcm}(a,b)=12xy=720\), so \(xy=60\). Since \(x\) and \(y\) are coprime, each prime power of \(60=2^2\cdot3\cdot5\) goes entirely to either \(x\) or \(y\). The unordered pairs \((x,y)\) are \((1,60)\), \((3,20)\), \((4,15)\), \((5,12)\). Hence the pairs \((a,b)\) are \((12,720)\), \((36,240)\), \((48,180)\), \((60,144)\).
A very useful model for problems with prescribed GCD and LCM.