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Korea 1999, problem 3
Posted:
Tue Apr 17, 2007 8:43 pm
by 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].
Posted:
Tue Apr 17, 2007 9:32 pm
by 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
Posted:
Wed Apr 18, 2007 11:11 am
by 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]