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(a) For all integers a, b, and c with a # 0, if a | b, then a | (bc).

(a) For all integers a, b, and c with a # 0, if a | b, then a | (bc).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer 3a

3. Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For all integers a, b, and c with a # 0, if a | b, then a | (bc). (b) For all integers a and b with a 0, if 6 | (ab), then 6 | a or 6 | b. (c) For all integers a, b, and c with a + 0, if a divides (b – 1) and a divides (c – 1), then a divides (bc – 1). - * (d) For each integer n, if 7 divides (n² – 4), then 7 divides (n – 2). * (e) For every integer n, 4n2 + 7n + 6 is an odd integer. (f) For every odd integer n, 4n2 + 7n + 6 is an odd integer. * (g) For all integers a, b, and d with d + 0, if d divides both a – b and a + b, then d divides a. (h) For all integers a, b, and c with a + 0, if a | (bc), then a | b or a | c.
Transcribed Image Text:3. Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For all integers a, b, and c with a # 0, if a | b, then a | (bc). (b) For all integers a and b with a 0, if 6 | (ab), then 6 | a or 6 | b. (c) For all integers a, b, and c with a + 0, if a divides (b – 1) and a divides (c – 1), then a divides (bc – 1). - * (d) For each integer n, if 7 divides (n² – 4), then 7 divides (n – 2). * (e) For every integer n, 4n2 + 7n + 6 is an odd integer. (f) For every odd integer n, 4n2 + 7n + 6 is an odd integer. * (g) For all integers a, b, and d with d + 0, if d divides both a – b and a + b, then d divides a. (h) For all integers a, b, and c with a + 0, if a | (bc), then a | b or a | c.
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