Multiple choice

There is a set of positive integers called Number Box which contains positive integers which when divided by 2, 3, 4, 5, 6 leave the remainders of 1, 2, 3, 4, 5 respectively. What is the maximum value of square root of the number of numbers that can be in the Number Box such that it does not contain any number with 4 digits?

  1. 4

  2. 8

  3. 5

  4. 3
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A Correct answer
Explanation

The number N satisfies N = LCM(2,3,4,5,6) * k - 1 = 60k - 1. We need 60k - 1 < 1000. 60k < 1001, k < 16.68. There are 16 such numbers (k=1 to 16). The square root of 16 is 4.

AI explanation

The numbers leave a remainder one less than the divisor in each case, meaning the numbers are of the form 12x minus 1, or equivalently, the least common multiple of 2, 3, 4, 5, and 6 is 60, so the numbers are 60x minus 1. The largest three-digit number of this form is 60 times 16 minus 1, which is 959, and the smallest is 60 times 1 minus 1, which is 59, giving 16 such numbers in the set. The maximum value of the square root of 16 is 4.

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