LCM of two numbers is 56 times their HCF , with the sum of their HCF and LCM being 1,710. If one of the two numbers is 240, then what is the other number?

The correct option is 210.

Let the Highest Common Factor (HCF) of the two numbers be $H$ and their Least Common Multiple (LCM) be $L$.

According to the problem, the LCM of the two numbers is 56 times their HCF:
$L=56H$

We are also given that the sum of their HCF and LCM is 1,710:
$L+H=1710$

Substitute the value of $L$ from the first equation into the second equation:
$56H+H=1710$

$57H=1710$

Solving for $H$:
$H=\frac{1710}{57}=30$

Now, calculate the LCM ($L$):
$L=56\times 30=1680$

We know the fundamental relationship between two numbers, their HCF, and their LCM:
$\text{Product of two numbers}=\text{HCF}\times \text{LCM}$

Let one number be 240 and the other number be $x$:
$240\times x=30\times 1680$

Solve for $x$:
$x=\frac{30\times 1680}{240}$

$x=\frac{1680}{8}=210$

Therefore, the other number is 210.