Multiple choice

In dividing a number by $585$, a student employed the method of short division. He divided the number successively by $5,9$ and $13$ (factors $585$) and got the remainders $4,8,12$ respectively. If he had divided the number by $585$, the remainder would have been.

  1. $24$
  2. $144$
  3. $292$
  4. $584$
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D Correct answer
Explanation

Using the Chinese Remainder Theorem or successive division logic, the number N = 585k + R. The remainders 4, 8, 12 from 5, 9, 13 imply N = 584.

AI explanation

Trace the successive short division backwards starting with a final quotient of 0. Before dividing by 13, the quotient was 13 multiplied by 0 plus 12, giving 12. Before dividing by 9, the quotient was 9 multiplied by 12 plus 8, resulting in 116. The original number is therefore 5 multiplied by 116 plus 4, which equals 584. Dividing 584 by 585 leaves a remainder of 584.

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