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Share-Based Compensation (Narrative) (Details) - USD ($)
$ in Millions
3 Months Ended 6 Months Ended
Feb. 28, 2025
Feb. 29, 2024
Feb. 28, 2025
Feb. 29, 2024
Share-Based Compensation Arrangement by Share-Based Payment Award [Line Items]        
Share-based compensation expense $ 2.1 $ 1.8 $ 4.1 $ 3.5