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Changes in Stockholders' Equity (Details) - USD ($)
$ / shares in Units, shares in Millions, $ in Millions
 3 Months Ended
Nov. 30, 2022
Nov. 30, 2021
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance $ 1,911.8 $ 2,044.5
Net income 74.9 87.6
Other comprehensive loss (0.4) (10.7)
Share-based payment amortization, issuances, and cancellations (1.8) 0.4
Employee stock purchase plan issuances 0.5 0.6
Cash dividends of $0.13 per share paid on common stock (4.3) (4.7)
Stock options exercised 0.4 8.0
Repurchases of common stock (77.6) (52.8)
Ending balance $ 1,903.5 $ 2,072.9
Cash dividends paid (usd per share) $ 0.13 $ 0.13
Common Stock Outstanding    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance (shares) 32.5 35.2
Beginning balance $ 0.5 $ 0.5
Share-based payment amortization, issuances, and cancellations (shares) 0.2 0.1
Stock options exercised (shares) 0.0 0.1
Repurchases of common stock (shares) (0.5) (0.3)
Ending balance (shares) 32.2 35.1
Ending balance $ 0.5 $ 0.5
Paid-in Capital    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance 1,036.3 995.6
Share-based payment amortization, issuances, and cancellations (1.8) 0.4
Employee stock purchase plan issuances 0.5 0.6
Stock options exercised 0.4 8.0
Ending balance 1,035.4 1,004.6
Retained Earnings    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance 3,176.2 2,810.3
Net income 74.9 87.6
Cash dividends of $0.13 per share paid on common stock (4.3) (4.7)
Ending balance 3,246.8 2,893.2
Accumulated Other Comprehensive Loss    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance (125.8) (98.2)
Other comprehensive loss (0.4) (10.7)
Ending balance (126.2) (108.9)
Treasury Stock, at cost    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance (2,175.4) (1,663.7)
Repurchases of common stock (77.6) (52.8)
Ending balance $ (2,253.0) $ (1,716.5)