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
.