A partial solution was  submitted by Ray Kremer.  From outside Bradley University, correct solutions were submitted by Steve Prowse, Ron Welch, Aaron David Kahn, Monty Gray, Lorenzo Pozzoli, Emanuele Macri, Massimo Brignone, Philippe Fondanaiche, Jan Siwanowicz, G. Simic.

This is true.   Let the n consecutive numbers be a+1, a+2,..., a+n.  The number of ways of choosing n students from a class of a+n students is (a+n)!/(a! n!) which is an integer, and solves the problem.

Mahendra Nambiar gave a beautifully simple proof by induction.

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ã2000 Alberto L. Delgado