The greatest number that divides $1005$, $1245$ and $1485$ leaving the same remainder is:

To find the greatest number that divides the given numbers leaving the same remainder, we calculate the Highest Common Factor (HCF) of the absolute differences between the numbers:
$$\text{Difference 1}=|1245-1005|=240$$

$$\text{Difference 2}=|1485-1245|=240$$

$$\text{Difference 3}=|1485-1005|=480$$

Now, we compute the $\text{HCF}(240,240,480)$:

* $240=240\times 1$
* $480=240\times 2$

Thus, the $\text{HCF}$ is $240$.