Changes in Stockholders' Equity (Details) - USD ($) $ / shares in Units, shares in Millions, $ in Millions	3 Months Ended
	Nov. 30, 2020	Nov. 30, 2019
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	$ 2,127.5	$ 1,918.9
Net income	59.6	57.0
Other comprehensive income (loss)	6.2	3.8
Share-based payment amortization, issuances, and cancellations	4.7	12.6
Employee stock purchase plan issuances	0.3	0.2
Cash dividends of $0.13 per share paid on common stock	(5.0)	(5.2)
Repurchases of common stock	(256.1)
Ending balance	$ 1,937.0	$ 1,987.3
Cash dividends paid (usd per share)	$ 0.13	$ 0.13
Common Stock Outstanding
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	$ 0.5	$ 0.5
Beginning balance (shares)	38.9	39.5
Share-based payment amortization, issuances, and cancellations (shares)	0.1	0.0
Repurchases of common stock (shares)	(2.6)
Ending balance	$ 0.5	$ 0.5
Ending balance (shares)	36.4	39.5
Paid-in Capital
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	$ 963.6	$ 930.0
Share-based payment amortization, issuances, and cancellations	4.7	12.6
Employee stock purchase plan issuances	0.3	0.2
Ending balance	968.6	942.8
Retained Earnings
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	2,523.3	2,295.8
Net income	59.6	57.0
Cash dividends of $0.13 per share paid on common stock	(5.0)	(5.2)
Ending balance	2,577.7	2,347.6
Accumulated Other Comprehensive Loss
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	(132.7)	(151.4)
Other comprehensive income (loss)	6.2	3.8
Ending balance	(126.5)	(147.6)
Treasury Stock, at cost
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	(1,227.2)	(1,156.0)
Repurchases of common stock	(256.1)
Ending balance	(1,483.3)	$ (1,156.0)
Cumulative effect of adoption of ASC 326
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	(0.2)
Cumulative effect of adoption of ASC 326 | Retained Earnings
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	$ (0.2)