The answer is 20.

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$2000|a^{b}b^{a}\Rightarrow 2|a \text{ or } b \text{ and } 5|a \text{ or } b. \\ \text{Thus, the least possible value of ab}=10. \\ \text{Now possibilities of (a,b)}=(1,10)(2,5)(5,2)(10,1). \\ \text{But in all these cases } 2000 \text{ doesn't divide } a^{b}b^{a}.\\ \text{The next multiple of }10 \text{ is } 20. \\ \text{Now } 2000|a^{b}b^{a} \text{ in some cases,} \\ \text{Answer}=\boxed{20}$