The greatest number that divides $415$, $595$ and $775$ leaving the same remainder is:

To find the greatest number that divides the numbers $N_1,N_2,N_3$ leaving the same remainder, we find the Highest Common Factor (HCF) of the absolute differences between the numbers: $|N_2-N_1|$, $|N_3-N_2|$, and $|N_3-N_1|$.
Given values: $415,595,775$

1. Calculate differences:
* $595-415=180$
* $775-595=180$
* $775-415=360$

2. Find the HCF of $180,180,$ and $360$:
* $180=180\times 1$
* $360=180\times 2$

The HCF is clearly $180$.