A 3-digit number N leaves the same remainder when dividing 2272 and 875. Find the sum of the digits of N.

Given data

• Remainders on dividing 2272 and 875 by the same 3-digit N are equal.

Concept / Approach

• If two numbers leave the same remainder upon division by N, then N divides their difference.

Step-by-step

Difference = 2272 − 875 = 1397So N must be a 3-digit divisor of 1397.Factor 1397: 1397 = 11 × 127The only 3-digit divisor is N = 127Sum of digits of N = 1 + 2 + 7 = 10

Verification

2272 mod 127 = 112; 875 mod 127 = 112. Remainders match.

Common pitfalls

• Choosing 11 which is not 3-digit.
• Not using the equal-remainder property to pass to a difference.

Final Answer

Sum of digits = 10.