Find the greatest number which when divided by $310$, $550$, and $728$ leaves the same remainder.

To find the greatest number that leaves the same remainder when dividing $310$, $550$, and $728$, we need to find the HCF of the absolute differences between the numbers.
Let the numbers be $a=310$, $b=550$, and $c=728$.
Differences:

$$|b-a|=550-310=240$$

$$|c-b|=728-550=178$$

$$|c-a|=728-310=418$$

Now, find the HCF of $240$, $178$, and $418$:

* $240=2^4\times 3\times 5$
* $178=2\times 89$
* $418=2\times 11\times 19$

The only common factor among all three differences is $2$.
Therefore, the greatest number is $2$.