# 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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