College Algebra Exam Review 26

College Algebra Exam Review 26 - Appendix C.) 1.6.8. Show...

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36 1. ALGEBRAIC THEMES 1.6.2. Prove Proposition 1.6.9 . Hint: For part (b), you have to show that if y 2 I.m;n/ , then xy 2 I.m:n/ . For part (c), you have to show that if y 2 I.m;n/ , then b divides y . 1.6.3. Suppose that a natural number p > 1 has the property that for all nonzero integers a and b , if p divides the product ab , then p divides a or p divides b . Show that p is prime. This is the converse of Proposition 1.6.16 . 1.6.4. For each of the following pairs of numbers m; n , compute g : c : d :.m;n/ and write g : c : d :.m;n/ explicitly as an integer linear combi- nation of m and n . (a) m D 60 and n D 8 (b) m D 32242 and n D 42 1.6.5. Show that for nonzero integers m and n , g : c : d :.m;n/ D g : c : d :. j m j ; j n j / . 1.6.6. Show that for nonzero integers m and n , g : c : d :.m;n/ is the largest natural number dividing m and n . 1.6.7. Show that for nonzero integers m and n , g : c : d :.m;n/ is the smallest element of I.m;n/ \ N . (The existence of a smallest element of I.m;n/ \ N follows from the well–ordering principle for the natural numbers; see
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Unformatted text preview: Appendix C.) 1.6.8. Show that two nonzero integers m and n are relatively prime if, and only if, 1 2 I.m;n/ . 1.6.9. Show that if p is a prime number and a is any nonzero integer, then either p divides a or p and a are relatively prime. 1.6.10. Suppose that a and b are relatively prime integers and that x is an integer. Show that if a divides the product bx , then a divides x . Hint: Use the existence of s;t such that sa C tb D 1 . 1.6.11. Suppose that a and b are relatively prime integers and that x is an integer. Show that if a divides x and b divides x , then ab divides x . 1.6.12. Show that if a prime number p divides a product a 1 a 2 :::a r of nonzero integers, then p divides one of the factors....
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