Difference between revisions of "2002 Indonesia MO Problems/Problem 1"

Revision as of 12:10, 14 July 2018

Problem

Show that $n^4 - n^2$ is divisible by $12$ for any integers $n > 1$ .

Solution

In order for $n^4 - n^2$ to be divisible by $12$ , it must be divisible by $4$ and $3$ . Note that $n^4 - n^2$ can be factored into $n^2 (n+1)(n-1)$ .

If $n$ is even, then $n^2 \equiv 0 \pmod{4}$ . If $n \equiv 1 \pmod{4}$ , then $n-1 \equiv 0 \pmod{4}$ , and if $n \equiv 3 \pmod{4}$ , then $n+1 \equiv 0 \pmod{4}$ . That means for all positive $n$ , $n^4 - n^2$ is divisible by $4$ .

If $n \equiv 0 \pmod{3}$ , then $n^2 \equiv 0 \pmod{3}$ . If $n \equiv 1 \pmod{3}$ , then $n-1 \equiv 0 \pmod{3}$ . If $n \equiv 2 \pmod{3}$ , then $n+1 \equiv 0 \pmod{3}$ . That means for all positive $n$ , $n^4 - n^2$ is divisible by $3$ .

Because $n^4 - n^2$ is divisible by $4$ and $3$ , it must be divisible by $12$ .

Revision as of 12:09, 14 July 2018 (view source) Rockmanex3 (talk \| contribs) (Solution to Problem 1)		Revision as of 12:10, 14 July 2018 (view source) Rockmanex3 (talk \| contribs) (→‎See Also) Newer edit →
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−	[[Category:~~Intermediate~~ Number Theory Problems]]	+	[[Category:Introductory Number Theory Problems]]

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Difference between revisions of "2002 Indonesia MO Problems/Problem 1"

Revision as of 12:10, 14 July 2018

Problem

Solution

See Also