Inclusion-Exclusion Princple

Question

Prove that for each 201 number that we will choose from [1,300] there will be always some x,y that thiers division will be power of 3.

For example:

$18 / 6$

Answer 1

Suppose you had $201$ distinct integers in $[1,300]$ such that none is a power of $3$ times another. If the least of these $\le 100$, multiply it by $3$ and the property is still satisfied. Repeat until there are no more numbers $\le 100$. But there are only $200$ integers from $101$ to $300$.