The purpose of this paper is to determine the number of fuzzy subgroups of a finite abelian group of order $p^{n}q^{m}$. As an application of our main result, explicit formulas for the number of fuzzy subgroups of $\mathbb{Z}_{p}^{n}\times\mathbb{Z}_{q}^{m}$ and $\mathbb{Z}_{p^{n}}\times\mathbb{Z}_{q}^{m}$ are given.
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