L26_Wed_Mar_24 - MAE3811Sp. 2010 Mahoney1 From Last time....

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1 MAE3811–Sp. 2010 Mahoney–1 From Last time…. Division In Depth Recall, it was recent, that for any integers a and b , with b not equal to 0 , a ÷ b = c if and only if c is the unique whole number such that b∙c = a . We say b divides a , written b | a , if, and only if, there is a unique integer q such that a = bq . If b | a , then b is a ______________ or a __________________ of a . a is a ___________________ of b . q usually stands for the ___________________ of a and b True or False (a - b) | a 2 –b 2 , a not equal to b 0 | 0 MAE3811–Sp. 2010 Mahoney–2 Properties of divisibility Recall a | b means…. Theorem: a | b if and only if a | -b Proof: Theorem: if d | a, and n is any integer, then d | na Proof:
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2 MAE3811–Sp. 2010 Mahoney–3 Properties of divisibility II Let a, b, and d be any integers with d ≠ 0. Theorem: If d | a and d | b, then d | (a + b). Proof: Corollary: d | a and d | b, then d | (a - b) Proof: MAE3811–Sp. 2010 Mahoney–4 Proof by Contradiction Proof by contradiction is a method of Proof often used when you wish to prove the
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L26_Wed_Mar_24 - MAE3811Sp. 2010 Mahoney1 From Last time....

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