  We prove limit theorems for the greatest common divisor and the least common
multiple of random integers. While the case of integers uniformly distributed
on a hypercube with growing size is classical, we look at the uniform
distribution on sublevel sets of multivariate symmetric polynomials, which we
call hyperbolic regions. Along the way of deriving our main results, we obtain
some asymptotic estimates for the number of integer points in these hyperbolic
domains, when their size goes to infinity.
