An olympiad adventure

Find the least positive integer \(p\) for which there exists a set \(\{\lambda_1,\lambda_2,\ldots,\lambda_p\}\) consisting of \(p\) distinct positive integers such that \[\left( 1-\frac{1}{\lambda_1} \right) \left( 1-\frac{1}{\lambda_2} \right) \cdots \left( 1-\frac{1}{\lambda_p}\right)=\frac{51}{2010}\]

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