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Research on stock synchronization has always been a topic of concern to scholars and investors. In the past, the focus was mainly on equity concentration, foreign shareholding, audit quality, and other issues, not including indexes. This article uses the monthly data of the Taiwan Stock Exchange Capital Weighted Stock Index (TAIEX) to solve the problem of the index and stock synchronization. And use the technical theory of the gray system to solve the small sample and uncertain problem. The discovery of the synchronization between these indexes and stock prices may provide investors with sufficient reference to make investment decisions.