2002 Indonesia MO Problems/Problem 1
Show that is divisible by for any integers .
In order for to be divisible by , must be divisible by and .
Lemma 1: is divisible by 4
Note that can be factored into . If is even, then . If , then , and if , then . That means for all positive , is divisible by .
Lemma 2: is divisible by 3
Again, note that can be factored into . If , then . If , then . If , then . That means for all positive , is divisible by .
Because is divisible by and , must be divisible by .
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