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, 2023	Feb. 28, 2022	Feb. 28, 2023	Feb. 28, 2022
Numerator
Net income used for calculating Basic EPS	$ 131,593	$ 109,938	$ 268,391	$ 217,585
Net income used for calculating Diluted EPS	$ 131,593	$ 109,938	$ 268,391	$ 217,585
Denominator
Weighted average common shares used in the calculation of Basic EPS (in shares)	38,281	37,837	38,201	37,685
Common stock equivalents associated with stock-based compensation plan (in shares)	700	924	746	943
Shares used in the calculation of Diluted EPS (in shares)	38,981	38,761	38,947	38,628
Basic EPS (in USD per share)	$ 3.44	$ 2.91	$ 7.03	$ 5.77
Diluted EPS (in USD per share)	$ 3.38	$ 2.84	$ 6.89	$ 5.63