Income Per Share - Schedule of Computation of Diluted Income Per Share (Details) - USD ($) $ / shares in Units, shares in Thousands, $ in Thousands	3 Months Ended
	Nov. 30, 2024	Nov. 25, 2023
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 43,105	$ 42,325
As reported - Basic (in shares)	18,586	18,702
As reported - Diluted	$ 43,105	$ 42,325
As reported - Diluted (in shares)	18,666	18,769
As reported - Diluted (in dollars per share)	$ 2.31	$ 2.26
Common Stock
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 36,213	$ 35,566
As reported - Basic (in shares)	15,012	15,111
As reported - Basic (in dollars per share)	$ 2.41	$ 2.35
Add: effect of dilutive potential common shares	$ 0	$ 0
Add: effect of dilutive potential common shares (in shares)	80	68
As reported - Diluted (in dollars per share)	$ 2.31	$ 2.26
Class B Common Stock
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 6,892	$ 6,759
As reported - Basic (in shares)	3,574	3,590
As reported - Basic (in dollars per share)	$ 1.93	$ 1.88