The number 2272 and 875 are divided by a 3 digit number N, giving same remainders. The sum of digit is :

Let the remainder in each case be x
Then, (2272 - x) and (875 - x) are exactly divisible by three digit number
Difference :
= (2272 - x) - (875 - x)
= 1397
Factor of 1397 = 11 × 127
Since, both 11 and 127 are prime number
Three digit number is 127
Sum of digits = 1 + 2 + 7 = 10
Note: - In this type of questions the number which divide the given number and leaves no remainder is either difference of number or a factor of difference.