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Changes in Stockholders' Equity (Details) - USD ($)
$ / shares in Units, shares in Millions, $ in Millions
 3 Months Ended 6 Months Ended
Feb. 29, 2024
Nov. 30, 2023
Feb. 28, 2023
Nov. 30, 2022
Feb. 29, 2024
Feb. 28, 2023
Increase (Decrease) in Stockholders' Equity [Roll Forward]            
Beginning balance $ 2,064.1 $ 2,015.4 $ 1,903.5 $ 1,911.8 $ 2,015.4 $ 1,911.8
Net income 89.2 100.6 83.2 74.9 189.8 158.1
Other comprehensive (loss) income 1.4 (1.5) (0.1) (0.4) (0.1) (0.5)
Share-based payment amortization, issuances, and cancellations 11.8 2.1 10.9 (1.8)    
Employee stock purchase plan issuances 0.3 0.5 0.3 0.5    
Cash dividends of $0.13 per share paid on common stock (4.7) (4.1) (4.2) (4.3)    
Stock options exercised 5.1 1.1 0.5 0.4    
Repurchases of common stock (17.6) (50.0) (46.5) (77.6)    
Ending balance $ 2,149.6 $ 2,064.1 $ 1,947.6 $ 1,903.5 $ 2,149.6 $ 1,947.6
Cash dividends paid (in usd per share) $ 0.15 $ 0.13 $ 0.13 $ 0.13    
Common Stock Outstanding            
Increase (Decrease) in Stockholders' Equity [Roll Forward]            
Beginning balance (in shares) 30.9 31.1 32.2 32.5 31.1 32.5
Beginning balance $ 0.5 $ 0.5 $ 0.5 $ 0.5 $ 0.5 $ 0.5
Share-based payment amortization, issuances, and cancellations (in shares)   0.1   0.2    
Repurchases of common stock (shares) (0.1) (0.3) (0.2) (0.5)    
Ending balance (in shares) 30.8 30.9 32.0 32.2 30.8 32.0
Ending balance $ 0.5 $ 0.5 $ 0.5 $ 0.5 $ 0.5 $ 0.5
Paid-in Capital            
Increase (Decrease) in Stockholders' Equity [Roll Forward]            
Beginning balance 1,070.5 1,066.8 1,035.4 1,036.3 1,066.8 1,036.3
Share-based payment amortization, issuances, and cancellations 11.8 2.1 10.9 (1.8)    
Employee stock purchase plan issuances 0.3 0.5 0.3 0.5    
Stock options exercised 5.1 1.1 0.5 0.4    
Ending balance 1,087.7 1,070.5 1,047.1 1,035.4 1,087.7 1,047.1
Retained Earnings            
Increase (Decrease) in Stockholders' Equity [Roll Forward]            
Beginning balance 3,601.9 3,505.4 3,246.8 3,176.2 3,505.4 3,176.2
Net income 89.2 100.6 83.2 74.9    
Cash dividends of $0.13 per share paid on common stock (4.7) (4.1) (4.2) (4.3)    
Ending balance 3,686.4 3,601.9 3,325.8 3,246.8 3,686.4 3,325.8
Accumulated Other Comprehensive Loss            
Increase (Decrease) in Stockholders' Equity [Roll Forward]            
Beginning balance (114.1) (112.6) (126.2) (125.8) (112.6) (125.8)
Other comprehensive (loss) income 1.4 (1.5) (0.1) (0.4)    
Ending balance (112.7) (114.1) (126.3) (126.2) (112.7) (126.3)
Treasury Stock, at cost            
Increase (Decrease) in Stockholders' Equity [Roll Forward]            
Beginning balance (2,494.7) (2,444.7) (2,253.0) (2,175.4) (2,444.7) (2,175.4)
Repurchases of common stock (17.6) (50.0) (46.5) (77.6)    
Ending balance $ (2,512.3) $ (2,494.7) $ (2,299.5) $ (2,253.0) $ (2,512.3) $ (2,299.5)