Also, regarding the previously mentioned n, I wanted to add that the previously mentioned \det(A^\top A) (a.k.a. product of singular values) is indeed a valid n when A^\top A has full column rank. If A^\top A does not have full column rank, however, then \det(A^\top A)=0, which you might agree is not what we’re looking for — we want something strictly positive! However, in this case, we can still take the product of the nonzero singular values. Denoting this quantity (the product of the nonzero singular values) by \text{volvol}(A) (“squared volume”), we can show that we again have n\mid\text{volvol}(A).
(This post and the above post are a slightly modified version of a discussion at Math.StackExchange.)