2000 IMO Problems/Problem 5
Does there exist a positive integer such that has exactly 2000 prime divisors and divides ?
Let . We will assume for the sake of contradiction that .
(mod ) (mod ). So 2 does not divide , and so is odd.
Select an arbitrary prime factor of and call it . Let's represent in the form , where is not divisible by .
Note that and are both odd since is odd. By repeated applications of Fermat's Little Theorem:
Continuing in this manner, and inducting on k from 1 to ,
(mod ) (mod )
So we have (mod )
Since is relatively prime to , (mod ) (mod )
Since is odd, is not divisible by . Hence is not divisible by . So we have a contradiction, and our original assumption was false, and therefore is still not divisible by .