Suppose you arbitrarily select 101 numbers from 1,2,3,...200. Show that there exists 2 numbers among the 101 chosen numbers such that one number is divisible by the other.

Group the 101 selected numbers by largest odd divisor. There are only 100 possible groups (1,3,5,...,199) so one group must have two elements. These two numbers' factorizations differ only by a power of two, so one divides the other.

Group the 101 selected numbers by largest odd divisor. There are only 100 possible groups (1,3,5,...,199) so one group must have two elements. These two numbers' factorizations differ only by a power of two, so one divides the other.

http://en.wikipedia.org/wiki/Abouabdillah%27s_theorem#Number_theory