Question

After the division of a number successively by 3, 4 and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divides the same number?

• A. 80
• B. 75
• C. 41
• D. 53

Hint:

For problems involving multiple divisibility and remainders, use the Chinese remainder theorem to find a solution.

Answer 1

The Correct Option is C

Use the Chinese remainder theorem or solve the system of congruences: \[ x \equiv 2 \mod 3, \quad x \equiv 1 \mod 4, \quad x \equiv 4 \mod 7. \] The solution to this system modulo 84 gives the remainder as 41.