Baltic Way 1993 · Problem 5

Solution:

Factorizing the expression, we get

$$n^{12} - n^{8} - n^{4} + 1 = (n^{4} + 1)(n^{2} + 1)^{2}(n - 1)^{2}(n + 1)^{2}.$$

Now note that one of the two even numbers $n - 1$ and $n + 1$ is divisible by $4$.