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)
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 55E: The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for...
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