Jure wrote positive integers 1 through 2015 on a whiteboard. Urška then inspected the written numbers from smallest to largest respectively and erased each number that was not divisible by 3. From the numbers still left on the whiteboard she then erased from smallest to largest each number that was not divisible by . From the remaining numbers she then erased from smallest to largest each number that was not divisible by , and so on. Which number did Urška erase last?
, 2015
Solution
After Urška inspected all the written numbers for the first time only numbers divisible by 3 were left on the whiteboard. After her second inspection only numbers divisible by were left. After the third inspection only numbers divisible by were left, and so on. The largest power of the number 3 smaller than 2015 is since . Thus, after Urška inspected the numbers for the sixth time only the numbers divisible by 729 were left on the whiteboard, they are 729 in 1458. In the next step she erased both this numbers since they are not divisible by 2187. Urška erased the number 1458 last.
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.