Earnings Per Share - Weighted Average Shares Outstanding (Details) - USD ($) $ / shares in Units, shares in Thousands, $ in Thousands	3 Months Ended
	Nov. 30, 2021	Nov. 30, 2020
Numerator
Net income used for calculating basic income per share	$ 107,647	$ 101,206
Net income used for calculating diluted income per share	$ 107,647	$ 101,206
Denominator
Weighted average common shares used in the calculation of basic income per share (in shares)	37,678	38,007
Common stock equivalents associated with stock-based compensation plans (in shares)	963	690
Shares used in calculation of diluted income per share (in shares)	38,641	38,697
Basic income per share (in USD per share)	$ 2.86	$ 2.66
Diluted income per share (in USD per share)	$ 2.79	$ 2.62