Finding numbers from LCM and HCF

Question

Let two numbers be $x$, $y$ and their LCM and HCF be $L$ and $H$ respectively. Then,

$7H=L$ and $x+y=392$ How can I find $x$ and $y$? I have tried to use $7 (\gcd)^2=xy$ but nothing came out of it. Any hints?

Answer 1

Let $x = a H, y = b H$, where $\gcd(a,b) = 1 $. Then $L = abH$, so $ab=7$. Thus, we must have $\{a, b \} = \{1, 7\}$. WLOG, $x=H, y = 7H$.

Answer 2

Hint:

$\gcd(x,y)=h \implies x=kh, y=mh$ and LCM=$kmh$, $7\gcd=LCM \implies km=7$