Let G be an almost simple group with socle A_n, the alternating group of degree n. We prove that there is a unit of order pq in the integral group ring of G if and only if there is an element of that order in G provided p and q are primes greater than n/3. We combine this with some explicit computations to verify the Prime Graph Question for all almost simple groups with socle A_n if n ≤ 17.