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Share-based Payments (Details) - USD ($)
$ in Millions
3 Months Ended 6 Months Ended
Feb. 28, 2022
Feb. 28, 2021
Feb. 28, 2022
Feb. 28, 2021
Share-based Payment Arrangement [Abstract]        
Share-based payment expense $ 10.0 $ 7.5 $ 17.6 $ 15.2