Consolidated Statements of Changes in Shareholders' Equity - USD ($) shares in Millions, $ in Millions	3 Months Ended		9 Months Ended		12 Months Ended
	May 27, 2023	May 28, 2022	May 27, 2023	May 28, 2022	Aug. 27, 2022
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	$ 1,337.3	$ 1,209.0	$ 1,263.0	$ 1,056.9	$ 1,056.9
Treasury stock, beginning balance (in shares)			(21.5)
Stock-based compensation	1.7	5.6	$ 8.2	12.5
Issuance of stock for employee benefit and stock-based awards, net	(0.1)		1.0	2.6
Issuance of stock for acquisition				22.0
Issuance of stock for settlement of earnout liability		13.2	0.0	13.2
Repurchase of common stock	(20.0)	(70.0)	(24.9)	(134.2)
Common stock dividends	(16.7)	(11.8)	(25.2)	(17.9)
Other		0.7		0.8
Total comprehensive income			0.1
Net income	59.1	117.2	172.1	308.0
Ending balance	$ 1,361.3	$ 1,263.9	$ 1,361.3	$ 1,263.9	$ 1,263.0
Treasury stock, ending balance (in shares)	(21.6)		(21.6)		(21.5)
Accounting standards update, extensible enumeration					Accounting Standards Update 2020-06
Adoption of Accounting Standards Update (ASU) 2020-06
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance			$ (33.0)
Ending balance					$ (33.0)
Common Shares
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance (in shares)	51.8	51.8	51.8	51.8	51.8
Beginning balance	$ 25.9	$ 25.9	$ 25.9	$ 25.9	$ 25.9
Ending balance (in shares)	51.8	51.8	51.8	51.8	51.8
Ending balance	$ 25.9	$ 25.9	$ 25.9	$ 25.9	$ 25.9
Additional Paid-In Capital
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	193.9	238.2	256.3	218.5	218.5
Stock-based compensation	1.7	5.6	8.2	12.5
Issuance of stock for employee benefit and stock-based awards, net	(0.1)		(7.0)	(1.9)
Issuance of stock for acquisition				14.7
Issuance of stock for settlement of earnout liability		7.8		7.8
Other		0.7		0.7
Ending balance	195.5	252.3	195.5	252.3	256.3
Additional Paid-In Capital | Adoption of Accounting Standards Update (ASU) 2020-06
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance			(62.0)
Ending balance					(62.0)
Retained Earnings
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	1,671.0	1,357.8	1,537.5	1,173.0	1,173.0
Common stock dividends	(16.7)	(11.8)	(25.2)	(17.9)
Other				0.1
Net income	59.1	117.2	172.1	308.0
Ending balance	1,713.4	1,463.2	1,713.4	1,463.2	1,537.5
Retained Earnings | Adoption of Accounting Standards Update (ASU) 2020-06
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance			29.0
Ending balance					29.0
Accumulated Other Comprehensive Income (Loss)
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	(0.4)	(0.5)	(0.5)	(0.5)	(0.5)
Total comprehensive income			0.1
Ending balance	(0.4)	(0.5)	(0.4)	(0.5)	(0.5)
Treasury Stock
Increase (Decrease) in Stockholders' Equity [Roll Forward]
Beginning balance	$ (553.1)	$ (412.4)	$ (556.2)	$ (360.0)	$ (360.0)
Treasury stock, beginning balance (in shares)	(21.2)	(19.0)	(21.5)	(18.7)	(18.7)
Issuance of stock for employee benefit and stock-based awards, net			$ 8.0	$ 4.5
Issuance of stock for employee benefit and stock-based awards, net (in shares)			0.4	0.2
Issuance of stock for acquisition				$ 7.3
Issuance of stock for acquisition (in shares)				0.4
Issuance of stock for settlement of earnout liability		$ 5.4		$ 5.4
Issuance of stock for settlement of earnout liability (in shares)		0.2		0.2
Repurchase of common stock (in shares)	(0.4)	(1.3)	(0.5)	(2.2)
Repurchase of common stock	$ (20.0)	$ (70.0)	$ (24.9)	$ (134.2)
Ending balance	$ (573.1)	$ (477.0)	$ (573.1)	$ (477.0)	$ (556.2)
Treasury stock, ending balance (in shares)	(21.6)	(20.1)	(21.6)	(20.1)	(21.5)