Which of the following numbers must be added to 5678 to give a remainder 35…

Concept: When a number N is divided by a divisor D, it leaves a certain remainder r₁. To make N (plus an added number x) leave a specific target remainder r₂, first find r₁, then compute x = r₂ − r₁, adjusted to lie between 0 and D − 1 by adding D if the raw difference is negative.

1. Divide 5678 by 460: 460 × 12 = 5520, so 5678 leaves remainder 5678 − 5520 = 158.

2. The required remainder after adding x is 35, so 158 + x must be congruent to 35 (mod 460).

3. This gives x = 35 − 158 = −123. Since a remainder cannot be negative, add the divisor 460: x = −123 + 460 = 337.

4. Hence, x = 337 must be added.

Cross-check: 5678 + 337 = 6015. Dividing 6015 by 460 gives 460 × 13 = 5980, leaving remainder 6015 − 5980 = 35, which matches the required condition.