Any integral circulant graph ICG(n,D) is characterised by its order n and a set D of positive divisors of n in such a way that it has vertex set Z/nZ and edge set {(a,b):a,b∈Z/nZ,gcd(a−b,n)∈D}. Such graphs are regular, and a connected ρ-regular graph G is called Ramanujan if the second largest modulus of the eigenvalues of the adjacency matrix of G is at most 2ρ−1.In 2010 Droll described all Ramanujan unitary Cayley graphs, i.e. graphs of type Xn:=ICG(n,{1}) having the Ramanujan property. Recently, Le and the author classified all Ramanujan graphs ICG(ps,D) for prime powers ps and arbitrary divisor sets D. We greatly extend the established results to graphs ICG(n,D) with arbitrary n and multiplicative divisor set D: (i) We derive a criterion (in terms of Euler's totient function) for ICG(n,D) to be Ramanujan. (ii) We prove that for all even integers n>2 and a positive proportion of the odd integers n, namely those having a “dominating” prime power factor, there exists a multiplicative divisor set D such that ICG(n,D) is Ramanujan. (iii) We show that the set of odd n for which no Ramanujan graph ICG(n,D) with multiplicative divisor set D exists, viz. ultrafriable integers, has positive density as well. The proofs of (ii) and (iii) use methods from analytic number theory.