On dividing $2272$ as well as $875$ by three digit number $N$, we get same remainder. What is the sum of the digits of $N$?
I cannot start hit and trial method here, so how should I do this?
You should start by putting "into math" what you know, which is
$2272 = a N + X$
$875 = b N + X$
Here a,b are positive integers and X is the common remainder.
Substracting these two we get
$1397 = c N $
with c a positive integer. There are multiple ways to continue, but the easiest here would be to look at the prime factors of 1397 (because you know you can write 1397 as the product of two integers (c,N))