Multiple choice

A number was divided successively in order by $4,5$ and $6$. The remainders were respectively $2,3$ and $4$. The number is:

  1. $214$
  2. $476$
  3. $954$
  4. $1908$
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A Correct answer
Explanation

Successive division means N = 4 * q1 + 2, q1 = 5 * q2 + 3, and q2 = 6 * q3 + 4. Substituting the equations backward with the smallest non-negative quotient q3 = 0 gives q2 = 4, q1 = 23, and N = 94. The general solution is N = 94 + 120k for any non-negative integer k. For k = 1, we get N = 214, which is one of the options.

AI explanation

Work backwards through the successive division steps using the formula: Previous Quotient equals Multiplier multiplied by Current Quotient plus Current Remainder. Starting with a final quotient of 1, the quotient before division by 6 is 1 multiplied by 6 plus 4, giving 10. Continuing backward, the quotient before division by 5 is 5 multiplied by 10 plus 3, resulting in 53. Finally, the original number is 4 multiplied by 53 plus 2, which equals 214.

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