# 2000 IMO Problems/Problem 5

Does there exist a positive integer such that has exactly 2000 prime divisors and divides ?

## Solution

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:

(mod )

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 .