When a three-digit integer n divides the numbers 2455 and 2945, n has the same…

Same-remainder rule: if an integer n divides two numbers a and b and leaves the same remainder in both cases, then n must divide their difference (a − b). This is because if a = q1n + r and b = q2n + r for the same remainder r, then a − b = (q1 − q2)n, which is an exact multiple of n.

1. Compute the difference between the two numbers: 2945 − 2455 = 490.

2. Since n leaves the same remainder in both 2455 and 2945, n must divide 490 exactly (by the rule above).

3. Factorize 490: 490 = 2 × 5 × 72.

4. List the three-digit divisors of 490: 245 and 490 itself.

5. Only 245 appears among the given options, so n = 245.

Check: 2455 ÷ 245 = 10 remainder 5, and 2945 ÷ 245 = 12 remainder 5 — both give the same remainder, confirming n = 245 satisfies the condition.