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Earnings Per Share - Basic and Diluted Earnings Per Share, Treasury Stock Method (Details) - USD ($)
$ / shares in Units, shares in Thousands, $ in Millions
 3 Months Ended
Nov. 30, 2023
Nov. 30, 2022
Earnings Per Share [Abstract]    
Net income $ 100.6 $ 74.9
Basic weighted average shares outstanding (in shares) [1] 31,005 32,308
Common stock equivalents (in shares) 360 396
Diluted weighted average shares outstanding (in shares) [1] 31,365 32,704
Basic earnings per share (in dollars per share) [1] $ 3.25 $ 2.32
Diluted earnings per share (in dollars per share) [1] $ 3.21 $ 2.29
[1] Earnings per share is calculated using unrounded numbers. Amounts in the table may not recalculate exactly due to rounding.