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3. Find x and y such that 314x + 159y = 1. 4. Find x and y such that 4144x + 7696y = 592. 5. If N = abc + 1, prove that (N, a) = (N, b) = (N, c) = 1. 6. Find two different solutions of 299x + 247y = 13. 7. Prove that if a b and bla, then a = b or a = -b. 8. Prove that if a b and a > 0, then (a, b) 9. Prove that ((a, b), b) (a, b). 10. = 11. = a. a. Prove that (n, n + 1) = 1 for all n > 0. b. If n > 0, what can (n, n + 2) be? a. Prove that (k, n +k) 1 if and only if (k, n) = 1. b. Is it true that (k, n + k) = d if and only if (k, n) = d? 12. Prove: If a b and c|d, then ac bd. =

3. Find x and y such that 314x + 159y = 1. 4. Find x and y such that 4144x + 7696y = 592. 5. If N = abc + 1, prove that (N, a) = (N, b) = (N, c) = 1. 6. Find two different solutions of 299x + 247y = 13. 7. Prove that if a b and bla, then a = b or a = -b. 8. Prove that if a b and a > 0, then (a, b) 9. Prove that ((a, b), b) (a, b). 10. = 11. = a. a. Prove that (n, n + 1) = 1 for all n > 0. b. If n > 0, what can (n, n + 2) be? a. Prove that (k, n +k) 1 if and only if (k, n) = 1. b. Is it true that (k, n + k) = d if and only if (k, n) = d? 12. Prove: If a b and c|d, then ac bd. =

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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Problem 3 & 12

Problems2 1. Calculate (314, 159) and (4144, 7696). 2. Calculate (3141, 1592) and (10001, 100083). 3. Find x and y such that 314x + 159y = 1. = (N, b) (N, c) 4. Find x and y such that 4144x + 7696y = 592. 5. If N = abc + 1, prove that (N, a) 6. Find two different solutions of 299x + 247y = 13. 7. Prove that if a|b and b|a, then a = b or a = -b. 8. Prove that if a|b and a > 0, then (a, b) 9. Prove that ((a, b), b) = (a, b). 10. 11. = = a. a. Prove that (n, n + 1) 1 for all n > 0. = b. If n > 0, what can (n, n + 2) be? 12. Prove: If a b and c|d, then ac|bd. 13. Prove: If d|a and d|b, then d²|ab. 14. Prove: If cab and (c, a) = d, then c|db. 15. = 1. a. Prove that (k, n +k) = 1 if and only if (k, n) = 1. b. Is it true that (k, n + k) = d if and only if (k, n) = d? a. If x² + ax + b = 0 has an integer root, show that it divides b. b. If x² + ax + b = 0 has a rational root, show that it is in fact an integer.
Transcribed Image Text:Problems2 1. Calculate (314, 159) and (4144, 7696). 2. Calculate (3141, 1592) and (10001, 100083). 3. Find x and y such that 314x + 159y = 1. = (N, b) (N, c) 4. Find x and y such that 4144x + 7696y = 592. 5. If N = abc + 1, prove that (N, a) 6. Find two different solutions of 299x + 247y = 13. 7. Prove that if a|b and b|a, then a = b or a = -b. 8. Prove that if a|b and a > 0, then (a, b) 9. Prove that ((a, b), b) = (a, b). 10. 11. = = a. a. Prove that (n, n + 1) 1 for all n > 0. = b. If n > 0, what can (n, n + 2) be? 12. Prove: If a b and c|d, then ac|bd. 13. Prove: If d|a and d|b, then d²|ab. 14. Prove: If cab and (c, a) = d, then c|db. 15. = 1. a. Prove that (k, n +k) = 1 if and only if (k, n) = 1. b. Is it true that (k, n + k) = d if and only if (k, n) = d? a. If x² + ax + b = 0 has an integer root, show that it divides b. b. If x² + ax + b = 0 has a rational root, show that it is in fact an integer.
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