Which of the following numbers must be added to 2345 to give a remainder 55…

Concept: When a number N leaves remainder r when divided by d, adding some value x changes the remainder to (r + x) mod d. To make the sum leave a target remainder R, the added value must satisfy x ≡ (R − r) (mod d), taking the smallest non-negative value that fits among the given options.

Application:

1. Divide 2345 by 276: 276 × 8 = 2208, so 2345 leaves a remainder of 2345 − 2208 = 137.

2. The sum (2345 + x) must leave remainder 55 when divided by 276, so (137 + x) must be congruent to 55 (mod 276).

3. Solve for x: x ≡ 55 − 137 = −82 (mod 276). Adding 276 to make it non-negative: x ≡ 276 − 82 = 194.

4. So x = 194.

Cross-check: Adding 194 to 2345 gives 2539. Since 276 × 9 = 2484, the remainder is 2539 − 2484 = 55 — confirming the value 194 satisfies the condition.