ABC26GN0433 · H.C.F. and L.C.M. of Numbers

Let L and H denote the L.C.M and H.C.F of the two numbers. Then, $L+H=592$ …(i) And, $L-H=518$ …(ii) Adding (i) and (ii), we get: $2L=1110$ or $L=555$. $\therefore H=592-555=37$. So, H.C.F. = 37 and L.C.M. = 555. Let the numbers be $x$ and $(296-x)$. Then, $x(296-x)=555\times 37 \Rightarrow x^{2}-296x+20535=0 \Rightarrow x^{2}-185x-111x+20535=0 \Rightarrow x(x-185)-111(x-185)=0 \Rightarrow (x-185)(x-111)=0 \Rightarrow x=185$ or $x=111$. Hence the numbers are 111 and 185.