Earnings Per Share - Weighted Average Shares Outstanding (Details) - USD ($) $ / shares in Units, shares in Thousands, $ in Thousands	3 Months Ended		6 Months Ended
	Feb. 28, 2022	Feb. 28, 2021	Feb. 28, 2022	Feb. 28, 2021
Numerator
Net income used for calculating basic income per share	$ 109,938	$ 96,643	$ 217,585	$ 197,849
Net income used for calculating diluted income per share	$ 109,938	$ 96,643	$ 217,585	$ 197,849
Denominator
Weighted average common shares used in the calculation of basic income per share (in shares)	37,837	37,916	37,685	37,961
Common stock equivalents associated with stock-based compensation plans (in shares)	924	704	943	697
Shares used in calculation of diluted income per share (in shares)	38,761	38,620	38,628	38,658
Basic income per share (in USD per share)	$ 2.91	$ 2.55	$ 5.77	$ 5.21
Diluted income per share (in USD per share)	$ 2.84	$ 2.50	$ 5.63	$ 5.12