Question

The LCM of two numbers is 60. Their GCf is 6 and their difference is 18. What are the two numbers?

Answer 1

`LCM(x,y)=60\\ GCF(x,y)=6\\ x-y=18\\\\ LCM(x,y)=(|x\cdot y|)/(GCD(x,y))\\ 60=(|x\cdot y|)/(6)\\ |x\cdot y|=360\\\\\\ |x\cdot y|=360\\ x-y=18\\\\ |x\cdot y|=360\\ x=18+y\\\\ |(18+y)\cdot y|=360\\ |18y+y^2|=360\\ 18y+y^2=360 \vee 18y+y^2=-360\\ y^2+18y+81=441 \vee y^2+18y+81=-279\\ (y+9)^2=441 \vee (y+9)^2=-279\\ y+9=√(441) \vee y+9=-√(441)\\ y=21-9 \vee y=-21-9\\ y=12 \vee y=-30\\\\ x-12=18 \vee x-(-30)=18\\ x=30 \vee x=-12\\\\ \boxed{x=30, y=12 \vee x=-12, y=-30}`