Too Many Variables!

Let \(A, B\) be sets of non-negative integers, each containing exactly 10 elements, satisfying the property that any integer between 1 and 100 inclusive can be expressed as the sum of an element in \(A\) and an element in \(B\); that is, for each \(n \in \{1, 2, \ldots, 100\}\), there exists \(a \in A\) and \(b \in B\) such that \(n = a+b\). How many unordered pairs of such \(\{A, B\}\) are there?

As an explicit example, this is one solution of the possible sets: \[\begin{eqnarray} A&=& \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \\ B&=&\{1, 11, 21, 31, 41, 51, 61, 71, 81, 91\} . \end{eqnarray} \] Also, note that \[\begin{eqnarray} A&=&\{1, 11, 21, 31, 41, 51, 61, 71, 81, 91\} \\ B&=&\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}. \end{eqnarray} \] counts as the same solution as above.

\[\] Clue: Cyclotomic polynomials.

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