The gcd of $x^n-1$ and $x^m-1$ is $x^{gcd(n,m)}-1$.
Is it known what the greatest common divisor of
$(x^{n_1}-1)(x^{n_2}-1)$ and $(x^{m_1}-1)(x^{m_2}-1)$ is?

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2

Look up cyclotomic polynomials.
– Gjergji ZaimiFeb 19 '12 at 13:34