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The set of degrees of maps D(M,N), where M,N are closed oriented n-manifolds, always contains 0 and the set of degrees of self-maps D(M) always contains 0 and 1. Also, if a,b\in D(M), then ab\in D(M); a set A\subseteq \mathbb {Z} so that ab\in A for each a,b\in A is called multiplicative. On the one hand, not every infinite set of integers (containing 0) is a mapping degree set (Neofytidis, Wang, and Wang [Bull. Lond. Math. Soc. 55 (2023), pp. 1700–1717]) and, on the other hand, every finite set of integers (containing 0) is the mapping degree set of some 3-manifolds (Costoya, Muñoz and Viruel [ Finite sets containing zero are mapping degree sets , arXiv: 2301.13719 ]). We show the following: Not every multiplicative set A containing 0,1 is a self-mapping degree set. For each n\in \mathbb {N} and k\geq 3, every D(M,N) for n-manifolds M and N is D(P,Q) for some (n+k)-manifolds P and Q. As a consequence of (ii) and Costoya, Muñoz and Viruel, every finite set of integers (containing 0) is the mapping degree set of some n-manifolds for all n\neq 1,2,4,5. [ABSTRACT FROM AUTHOR]