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Whcih of the following statements is/are true?

A : $N$ is divisible by 1999, where $N=1\times 3\times 5\times \dots\times 1995\times 1997 + 2\times 4\times 6\times 8\times \dots \times 1996\times 1998.$

B : $M$ is divisible by 2001, where $M=2\times 4\times 6\times 8\times \dots \times 1998\times 2000 - 1\times 3\times 5\times \dots\times 1997\times 1999.$