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Equity
6 Months Ended
Jun. 30, 2018
Equity [Abstract]  
Equity Equity

Share Repurchases. During the six months ended June 30, 2018, we repurchased (i) 12,588,800 shares of our class A ordinary shares at an average price per share of $30.15 and (ii) 29,342,800 shares of our class C ordinary shares at an average price per share of $30.97, for an aggregate purchase price of $1,288.3 million, including direct acquisition costs. At June 30, 2018, the remaining amount authorized for share repurchases was $783.9 million. On July 31, 2018, our board of directors authorized an additional $500.0 million for share repurchases.