For \(A = (2^{334})!\) and \(B = \displaystyle \prod_{n=0}^{100}(10^n)!\)

\((A^{1000} - 1) / (A - 1) = P \times B + Q\) where \(P\) and \(Q\) are non-negative integers and \(Q < B\)

\(Q = 1000000 \times R + S\) where \(R\) and \(S\) are non-negative integers and \(S < 1000000\)

Find \(S\)

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