1995 AIME Problem 11

A right rectangular prism P (i.e., a rectangular parallelepiped) has sides of integral length a, b, c, with a \leq b \leq c. A plane parallel to one of the faces of P cuts P into two prisms, one of which is similar to P, and both of which have nonzero volume. Given that b=1995, for how many ordered triples (a, b, c) does such a plane exist?