Which of the following numbers must be added to 3456 to give a remainder of 26…

Concept: If a number N leaves remainder r when divided by d, then N = d × k + r for some integer k (N is congruent to r modulo d). To find what value must be added to a number so the new sum leaves a specified remainder when divided by d, express the required sum as the nearest suitable multiple of d plus that remainder, then solve for the added value.

Application:

1. Let x be the number added to 3456 so that (3456 + x) divided by 280 leaves remainder 26.

2. This means 3456 + x = 280k + 26 for some integer k, i.e. 3430 + x = 280k.

3. So (3430 + x) must be a multiple of 280. Dividing 3430 by 280: 280 × 12 = 3360, remainder 70, so 3430 = 280 × 12 + 70.

4. For 3430 + x to become the next multiple of 280 (280 × 13 = 3640), x must make up the remaining 280 - 70 = 210.

5. So x = 210.

Cross-check: Adding 210 to 3456 gives 3666. Since 280 × 13 = 3640, dividing 3666 by 280 leaves remainder 3666 - 3640 = 26, matching the required remainder and confirming x = 210.