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Let a, b,c be non-zero integers. Suppose that x, y are integers such that ax + by = c. (i) Show that for any integer z, zb za (ii) Suppose that r,s are integers (possibly different from x,y respectively) such that ar + bs = c. Show that there exists some integer w such that wb r =x- wa gcd(a,b) and s= y+ ged(a, b)

Let a, b,c be non-zero integers. Suppose that x, y are integers such that ax + by = c. (i) Show that for any integer z, zb za (ii) Suppose that r,s are integers (possibly different from x,y respectively) such that ar + bs = c. Show that there exists some integer w such that wb r =x- wa gcd(a,b) and s= y+ ged(a, b)

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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plz provide ans fr part c

Question 3 (a) Define a sequence {an} as follows: Let az = 1, and define (an)? + (2n + 3)an + (4n + 3) an+1 an + 2 for n 2 1. Use Mathematical Induction to show that n? s an < (n+ 1)? for all positive integers n. (b) Use the Euclidean Algorithm to find d = ged(4753,1379) and express d as an integral linear combination of 4753 and 1379. (c) Let a, b,c be non-zero integers. Suppose that x, y are integers such that ax + by = c. (i) Show that for any integer z, zb a(x- god(a.b) + b (y +pod(a.) = c. gcd(a, b) Suppose that r, s are integers (possibly different from x,y respectively) such that ar + bs = c. Show that there exists some integer w such that (ii) wb wa r =x- and s= y + gcd(a,b) gcd(a, b)
Transcribed Image Text:Question 3 (a) Define a sequence {an} as follows: Let az = 1, and define (an)? + (2n + 3)an + (4n + 3) an+1 an + 2 for n 2 1. Use Mathematical Induction to show that n? s an < (n+ 1)? for all positive integers n. (b) Use the Euclidean Algorithm to find d = ged(4753,1379) and express d as an integral linear combination of 4753 and 1379. (c) Let a, b,c be non-zero integers. Suppose that x, y are integers such that ax + by = c. (i) Show that for any integer z, zb a(x- god(a.b) + b (y +pod(a.) = c. gcd(a, b) Suppose that r, s are integers (possibly different from x,y respectively) such that ar + bs = c. Show that there exists some integer w such that (ii) wb wa r =x- and s= y + gcd(a,b) gcd(a, b)
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