Multiple choice

A number lying between $1000$ and $2000$ is such that on division by $2, 3, 4, 5, 6, 7$ and $8$ leaves remainders $1, 2, 3, 4, 5, 6$ and $7$, respectively. What is the number?

  1. $1679$
  2. $1579$
  3. $1779$
  4. $1879$
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A Correct answer
Explanation

The number leaves remainders 1, 2, 3, 4, 5, 6, 7 when divided by 2, 3, 4, 5, 6, 7, 8. This means the number is (LCM of 2-8) - 1. LCM(2,3,4,5,6,7,8) = 840. The number is of the form 840k - 1. For k=2, 840*2 - 1 = 1679.

AI explanation

Notice that the difference between each divisor and its corresponding remainder is 1 (for example, 2 minus 1 is 1, and 3 minus 2 is 1). This means the number must be one less than a multiple of the least common multiple of 2, 3, 4, 5, 6, 7, and 8. The least common multiple of these divisors is 840. The multiples of 840 are 840 and 1680, so subtracting 1 gives 839 and 1679. Because the number must lie between 1000 and 2000, the required number is 1680 minus 1, which equals 1679.

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