Korea 1999, problem 3

[zefuri]

Find all positive integers n such that [unparseable or potentially dangerous latex formula] is a multiple of [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] is a divisor of [unparseable or potentially dangerous latex formula] for some integer [unparseable or potentially dangerous latex formula].

[stupidityismygam]

welll
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

so [unparseable or potentially dangerous latex formula]
let n=2a
so, it can be written as [unparseable or potentially dangerous latex formula]

WTF: [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula]

or [unparseable or potentially dangerous latex formula], so [unparseable or potentially dangerous latex formula]

so [unparseable or potentially dangerous latex formula]
then let x=4b+1
so we have [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula]

i have this...i dont think it helps though

[zefuri]

Its not complete yet...

First we must show that [unparseable or potentially dangerous latex formula] must be even to satisfy [unparseable or potentially dangerous latex formula].

By Fermat's Theorem, we know

[unparseable or potentially dangerous latex formula]

Hence,
[unparseable or potentially dangerous latex formula] for some [unparseable or potentially dangerous latex formula] that is equal to[unparseable or potentially dangerous latex formula]

Since,
[unparseable or potentially dangerous latex formula]

Thus,
[unparseable or potentially dangerous latex formula]

It remains to be proven that [unparseable or potentially dangerous latex formula] must be of the form [unparseable or potentially dangerous latex formula]
We can split this into two cases.

Case 1: Let [unparseable or potentially dangerous latex formula] have a prime odd divisor [unparseable or potentially dangerous latex formula]

Since,
[unparseable or potentially dangerous latex formula] is a multiple of [unparseable or potentially dangerous latex formula], we know [unparseable or potentially dangerous latex formula] is of the form [unparseable or potentially dangerous latex formula]

Hence,
[unparseable or potentially dangerous latex formula]

which is a contradiction.

This is all i have but i can't find an elegant way why [unparseable or potentially dangerous latex formula]