A three-digit integer X divides both 34041 and 32506, leaving the same…

Concept: If an integer divides two numbers and leaves the same remainder in each case, it must divide their difference exactly — the remainder cancels out when one number is subtracted from the other.

1. Difference = 34041 − 32506 = 1535.

2. Factorize 1535: 1535 = 5 × 307 (307 is prime).

3. Divisors of 1535 are 1, 5, 307 and 1535.

4. Since X must be a three-digit integer, the only divisor that fits is 307.

Cross-check: 34041 ÷ 307 = 110 remainder 271, and 32506 ÷ 307 = 105 remainder 271 — the same remainder in both, confirming X = 307.