  Given a variety over $\mathbb{Q}$, we study the distribution of the number of
primes dividing the coordinates as we vary an integral point.
  Under suitable assumptions, we show that this has a multivariate normal
distribution. We generalise this to more general Weil divisors, where we obtain
a geometric interpretation of the covariance matrix.
  For our results we develop a version of the Erd\H{o}s-Kac theorem that
applies to fairly general integer sequences and does not require a positive
exponent of level of distribution.
