Numbers and boxes

The integers from 1 to 1000 (inclusive) are to be placed into boxes such that no box contains a number that is a multiple of another number in the same box.

What is the minimum number of boxes needed?


As an explicit example, the integers from 1 to 10 can be placed into 4 boxes while satisfying the condition:

  • Box 1 - \( \{ 1 \} \)
  • Box 2 - \( \{ 2, 3, 5, 7 \} \)
  • Box 3 - \( \{ 4, 6, 9 , 10\} \)
  • Box 4 - \( \{ 8 \} \)
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