The LCM of the two numbers is $9$ times their HCF. The sum of LCM and HCF is $500.$ Find their HCF.

Let the HCF of two numbers be $x.$​

Then, LCM $=9x$

​​According to the question,​​ we have

LCM $+$ HCF$=500$ [Given]

$\Rightarrow 9x+x=500$

$\Rightarrow 10x=500$

$\Rightarrow x=50$
Hence, the HCF of the two numbers is $50.$