  We prove a strong version of quantum ergodicity for linear hyperbolic maps of
the torus (``cat maps''). We show that there is a density one sequence of
integers so that as N tends to infinity along this sequence, all eigenfunctions
of the quantum propagator at inverse Planck constant N are uniformly
distributed.
  A key step in the argument is to show that for a hyperbolic matrix in the
modular group, there is a density one sequence of integers N for which its
order (or period) modulo N is somewhat larger than the square root of N.
