  We provide lower bounds for p-adic valuations of multisums of factorial
ratios which satisfy an Ap\'ery-like recurrence relation: these include
Ap\'ery, Domb, Franel numbers, the numbers of abelian squares over a finite
alphabet, and constant terms of powers of certain Laurent polynomials. In
particular, we prove Beukers' conjectures on the p-adic valuation of Ap\'ery
numbers. Furthermore, we give an effective criterion for a sequence of
factorial ratios to satisfy the p-Lucas property for almost all primes p.
