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Changes in Stockholders' Equity (Details) - USD ($)
$ / shares in Units, shares in Millions, $ in Millions
 3 Months Ended
Nov. 30, 2024
Nov. 30, 2023
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance $ 2,378.8 $ 2,015.4
Net income 106.7 100.6
Other comprehensive loss (16.8) (1.5)
Share-based payment amortization, issuances, and cancellations (11.0) 2.1
Employee stock purchase plan issuances 0.6 0.5
Cash dividends paid on common stock (4.5) (4.1)
Stock options exercised 15.0 1.1
Repurchases of common stock (5.4) (50.0)
Ending balance $ 2,463.4 $ 2,064.1
Cash dividends paid (in usd per share) $ 0.15 $ 0.13
Common Stock Outstanding    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance (in shares) 30.8 31.1
Beginning balance $ 0.5 $ 0.5
Share-based payment amortization, issuances, and cancellations (in shares) 0.1 0.1
Stock options exercised (in shares) 0.1  
Repurchases of common stock (in shares) (0.1) (0.3)
Ending balance (in shares) 31.0 30.9
Ending balance $ 0.5 $ 0.5
Repurchases of common stock (in shares) 0.1 0.3
Paid-in Capital    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance $ 1,115.9 $ 1,066.8
Share-based payment amortization, issuances, and cancellations (11.0) 2.1
Employee stock purchase plan issuances 0.6 0.5
Stock options exercised 15.0 1.1
Ending balance 1,120.5 1,070.5
Retained Earnings    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance 3,909.8 3,505.4
Net income 106.7 100.6
Cash dividends paid on common stock (4.5) (4.1)
Ending balance 4,012.0 3,601.9
Accumulated Other Comprehensive Loss    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance (114.9) (112.6)
Other comprehensive loss (16.8) (1.5)
Ending balance (131.7) (114.1)
Treasury Stock, at cost    
Increase (Decrease) in Stockholders' Equity [Roll Forward]    
Beginning balance (2,532.5) (2,444.7)
Repurchases of common stock (5.4) (50.0)
Ending balance $ (2,537.9) $ (2,494.7)