The sum of LCM and HCF of two numbers is 854. If the LCM is 60 times the HCF and one of the numbers is 70, then the other number is:

The correct option is 168.

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

According to the problem statement, we are given two conditions:

1. The sum of the LCM and the HCF is 854:
$L+H=854$

2. The LCM is 60 times the HCF:
$L=60\times H$

We can substitute the expression for $L$ from the second condition into the first equation:
$60\times H+H=854$
$61\times H=854$

Solving for $H$:
$H=\frac{854}{61}$
$H=14$

Now, we can find the value of $L$ by substituting $H=14$ back into the relation for LCM:
$L=60\times 14$
$L=840$

We are given that one of the numbers is 70. Let this number be $a=70$, and let the other number be $b$.

A fundamental property of any two positive integers is that the product of the two numbers is equal to the product of their HCF and LCM:
$a\times b=H\times L$

Substitute the known values into this formula:
$70\times b=14\times 840$

Solving for the other number $b$:
$b=\frac{14\times 840}{70}$
$b=14\times 12$
$b=168$

Thus, the other number is indeed 168.