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. 26, 2022	Nov. 27, 2021
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 33,957	$ 33,705
As reported - Basic (in shares)	18,672	18,872
As reported – Diluted	$ 33,957	$ 33,705
As reported - Diluted (in shares)	18,754	19,026
As reported - Diluted (in dollars per share)	$ 1.81	$ 1.77
Common Stock
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 28,525	$ 28,291
As reported - Basic (in shares)	15,082	15,229
As reported - Basic (in dollars per share)	$ 1.89	$ 1.86
Add: effect of dilutive potential common shares	$ 0	$ 0
As reported – Diluted	$ 33,957	$ 33,705
Add: effect of dilutive potential common shares (in shares)	82	154
Class B Common Stock
Earnings Per Share Diluted [Line Items]
As reported - Basic	$ 5,432	$ 5,414
As reported - Basic (in shares)	3,590	3,643
As reported - Basic (in dollars per share)	$ 1.51	$ 1.49