When 2272 and 875 are each divided by the same three digit number N, they leave the same remainder. What is the sum of the digits of N?

Introduction / Context:
This problem is a classic application of properties of remainders and divisors. When two numbers give the same remainder upon division by the same divisor, the difference between these numbers is exactly divisible by that divisor. Here, we are told the divisor is a three digit number and asked to find the sum of its digits. This type of question is very common in number system sections of aptitude tests.

Given Data / Assumptions:
Two numbers: 2272 and 875.Both leave the same remainder when divided by a three digit number N.N is a three digit positive integer.We want the sum of the digits of N.

Concept / Approach:
If two numbers a and b leave the same remainder when divided by N, then a − b is divisible by N. This is a direct consequence of the representation a = Nq1 + r and b = Nq2 + r. Subtracting these gives a − b = N(q1 − q2), showing that N divides a − b. Once we compute the difference, we find its factors and select the three digit factor as the required divisor N. Then we sum the digits of that divisor.

Step-by-Step Solution:
Step 1: Compute the difference between the two numbers: 2272 − 875.Step 2: Perform the subtraction: 2272 − 875 = 1397.Step 3: Since both numbers leave the same remainder on division by N, N must be a factor of 1397.Step 4: Factorize 1397 into prime factors. It is equal to 11 * 127.Step 5: The positive factors of 1397 are 1, 11, 127 and 1397.Step 6: We are told N is a three digit number, so N must be 127.Step 7: Add the digits of N: 1 + 2 + 7 = 10.

Verification / Alternative check:
To verify, divide both 2272 and 875 by 127. When 2272 is divided by 127, we get 127 * 17 = 2159 and a remainder of 2272 − 2159 = 113. When 875 is divided by 127, we get 127 * 6 = 762 and a remainder of 875 − 762 = 113. Both divisions produce the same remainder, confirming that 127 is indeed the divisor N. The sum of the digits is still 10.

Why Other Options Are Wrong:
If N had digit sum 11, 12 or 13, N would be some other number, but none of those candidate numbers would divide 1397 while remaining three digit in the specified way. Only 127 fits both the factor condition and the three digit requirement, so the other sums are based on incorrect divisors.

Common Pitfalls:
A common mistake is to forget that N must be three digit and accidentally choose the factor 11 or 1397. Another error is to assume the remainder is zero, but the question clearly mentions that both divisions leave the same remainder, not necessarily zero. Carefully applying the property of equal remainders and verifying the factorization avoids these issues.

Final Answer:
The sum of the digits of the three digit divisor N is 10.