Question

if the sum of squares of lcm and hcf of two positive numbers is 3609 and their lcm is 57 more than their hcf then the product of two numbers is​

Answer 1

$$\begin{gathered} \mathrm{Hi}, \\ \mathrm{Let}\mathrm{the}\mathrm{LCM}\mathrm{is}\mathrm{x}\mathrm{and}\mathrm{HCF}\mathrm{is}\mathrm{y}\mathrm{so} \\ {\mathrm{x}}^2+{\mathrm{y}}^2=3609 \\ \mathrm{x}=57+\mathrm{y} \\ \mathrm{putting}\mathrm{in}\mathrm{first}\mathrm{equation} \\ {\left(57+\mathrm{y}\right)}^2+{\mathrm{y}}^2=3609 \\ 3249+2{\mathrm{y}}^2+114\mathrm{y}=3609 \\ 2{\mathrm{y}}^2+114\mathrm{y}-360=0 \\ {\mathrm{y}}^2+57\mathrm{y}-180=0 \\ {\mathrm{y}}^2+60\mathrm{y}-3\mathrm{y}-180=0 \\ \mathrm{y}\left(\mathrm{y}+60\right)-3\left(\mathrm{y}+60\right)=0 \\ \left(\mathrm{y}+60\right)\left(\mathrm{y}-3\right)=0 \\ \mathrm{y}=3,-60 \\ \mathrm{so}\mathrm{x}=60,-3 \\ \mathrm{now}\mathrm{we}\mathrm{know}\mathrm{that}\mathrm{product}\mathrm{of}\mathrm{LCM}\mathrm{and}\mathrm{HCF}=\mathrm{product}\mathrm{of}\mathrm{numbers} \\ \mathrm{so}\mathrm{product}\mathrm{of}\mathrm{numbers}=3\times 60\mathrm{or}-3\times \left(-60\right) \\ \mathrm{so}180\mathrm{in}\mathrm{each}\mathrm{case} \end{gathered}$$