2002 Indonesia MO Problems/Problem 1

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


Lemma 1: $n^4 - n^2$ is divisible by 4
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 $n$, $n^2 (n+1)(n-1)$ is divisible by $4$.


Lemma 2: $n^4 - n^2$ is divisible by 3
Again, note that $n^4 - n^2$ can be factored into $n^2 (n+1)(n-1)$. 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 $n$, $n^2 (n+1)(n-1)$ is divisible by $3$.


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

See Also

2002 Indonesia MO (Problems)
Preceded by
First Problem
1 2 3 4 5 6 7 Followed by
Problem 2
All Indonesia MO Problems and Solutions
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