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

Compute the current remainder: 5678 − 460×12 = 5678 − 5520 = 158.

• We need a number x so that (158 + x) ≡ 35 (mod 460).

• Solve the congruence: x ≡ 35 − 158 ≡ −123 ≡ 337 (mod 460). The smallest positive solution is 337.

• Any x of the form 337 + 460k (k an integer ≥ 0) will give remainder 35. In particular, 797 = 337 + 460, so adding 797 works.

Answer: 797