Abstract
This paper is a presentation and discussion of two proofs of a theorem. The theorem is a statement about a specific type of sequence of inputs where all multiples of fixed denominations are accepted as inputs of an automaton. The theorem has interesting implications for accepted strings in a nite automaton in general setting. Here, we have examined different methods for proving the theorem. We present one analytic proof by using automata theory with graph theoretic concepts and another proof by utilizing the ordered partition of number theory. We also illustrate the results with the help of examples.
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