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Goodwill and Other Intangible Assets (Schedule of Intangible Assets and Weighted Average Remaining Life Finite and Infinite Excluding Goodwill) (Details) - USD ($) $ in Thousands | Aug. 31, 2023 | Aug. 31, 2022
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Finite And Indefinite Intangible Assets [Line Items] |  |
Definite lived intangible assets: weighted average years | 5 years 8 months 12 days | 2 years 8 months 12 days
Total, gross carrying amount | $ 66,627 | $ 54,892
Definite lived intangible assets: accumulated amortization | (38,908) | (36,684)
Tradenames [Member] |  |
Finite And Indefinite Intangible Assets [Line Items] |  |
Indefinite lived intangible assets: gross carrying amount | $ 11,939 | $ 11,162
Patents and Developed Technology [Member] |  |
Finite And Indefinite Intangible Assets [Line Items] |  |
Definite lived intangible assets: weighted average years | 3 years 1 month 6 days | 3 years 3 months 18 days
Definite lived intangible assets: gross carrying amount | $ 28,658 | $ 26,329
Definite lived intangible assets: accumulated amortization | $ (24,402) | $ (24,106)
Customer Relationships [Member] |  |
Finite And Indefinite Intangible Assets [Line Items] |  |
Definite lived intangible assets: weighted average years | 6 years 9 months 18 days | 2 years 1 month 6 days
Definite lived intangible assets: gross carrying amount | $ 26,030 | $ 17,401
Definite lived intangible assets: accumulated amortization | $ (14,506) | $ (12,578)

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