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Share-based Compensation - Narrative (Details) - USD ($)
$ / shares in Units, $ in Millions
12 Months Ended
Aug. 31, 2024
Aug. 26, 2023
Aug. 27, 2022
Oct. 31, 2023
Stock Appreciation Rights (SARs)        
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]        
Percentage vested after each anniversary of the grant date 20.00%      
Expiration period 10 years      
Weighted average fair values of Share-Based Awards granted (in dollars per share) $ 55.97 $ 60.38 $ 51.25  
Stock Appreciation Rights (SARs) | Cliff Vesting        
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]        
Period awards become fully vested or exercisable 5 years      
Share-Based Awards        
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]        
Total compensation cost not yet recognized related to non-vested awards $ 16.0      
Period for recognition   3 years 7 days    
Time-Based Restricted Stock Units        
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]        
Percentage vested after each anniversary of the grant date 20.00%      
Period awards become fully vested or exercisable 5 years      
Non-Employee Directors | Unrestricted Stock        
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]        
Shares granted (in shares) 1,475 1,575 1,025  
The 2023 Plan        
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]        
Number of shares reserved for issuance (in shares)       375,000
Shares available for future grants (in shares) 375,000      
The 2023 Plan | Non-Employee Directors | Stock Appreciation Rights (SARs)        
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]        
Shares granted (in shares) 5,004 6,864 9,750