When each of the numbers 2272 and 875 is divided by the same three digit number N, the remainders obtained are equal. What is the sum of the digits of this divisor N?

Introduction:
This question tests your understanding of remainders and the use of the highest common factor (HCF) when dealing with the same remainder condition. It is a classic number theory problem often seen in aptitude examinations.

Given Data / Assumptions:

• When 2272 is divided by N, the remainder is some value R.
• When 875 is divided by the same N, the remainder is also R.
• N is a three digit number.
• We must find the sum of the digits of N.

Concept / Approach:
If two numbers leave the same remainder when divided by N, then their difference is exactly divisible by N. That is, N divides (2272 − 875). Therefore, N must be a factor of this difference. Because N is also specified to be a three digit divisor, we can factor the difference and choose the appropriate factor that has three digits, then sum its digits.

Step-by-Step Solution:
Let the common remainder be R. Then 2272 = N * a + R and 875 = N * b + R for some integers a and b. Subtract: 2272 − 875 = N(a − b). Compute the difference: 2272 − 875 = 1397. Thus, N divides 1397 exactly. Now factor 1397 to find its divisors. Check small primes: 1397 is not divisible by 2, 3, or 5. Test 7: 7 × 199 = 1393, so not divisible by 7. Test 11: 11 × 127 = 1397, so 1397 = 11 × 127. The positive divisors are 1, 11, 127, and 1397. Among these, the only three digit divisor is N = 127. So N = 127. Sum of digits of N = 1 + 2 + 7 = 10.

Verification / Alternative check:
You can verify that both 2272 and 875 leave the same remainder when divided by 127. Compute 2272 ÷ 127 and 875 ÷ 127, find their remainders and confirm they match. Since 127 is a three digit divisor of 1397, it correctly satisfies the given condition.

Why Other Options Are Wrong:
Digit sums 11, 12, and 13 would correspond to different three digit numbers, none of which divide 1397 exactly. Because 127 is the only three digit factor of 1397, any other choice would contradict the divisibility requirement implied by the equal remainders.

Common Pitfalls:
Learners sometimes try to find N by trial dividing 2272 and 875 directly, which is inefficient and error prone. Others forget to use the key idea that equal remainders imply that the difference of the numbers is exactly divisible by the divisor. Always reduce such problems to a difference and then factor that difference.

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