This paper is concerned with the problem of classifying tuples of natural numbers a(1), ... , a(K) and b(1), ... , b(L) such that the ratio of factorials Pi(K)(i=1)(a(i)n)!/Pi(L)(j=1)(b(j)n)/! is an integer for all natural numbers n. A complete solution to this problem is only known in the case L - K = 1, due to the work of Bober building on an observation of Rodriguez Villegas, which relies on the Beukers-Heckman classification of algebraic hypergeometric functions. We provide an alternative proof of this result, which also makes progress on the general problem when L - K > 1.
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