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If a and b are non-zero integers, then there exist four integers h,k,r,s such that hs-kr =1,ak+bs=0

If a and b are non-zero integers, then there exist four integers h,k,r,s such that hs-kr =1,ak+bs=0

Grade:9

2 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
9 years ago
Hi


but h0 and k0 we have to find by hit and trial by lemma its sure that it will be found given condition is satisfied.
Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty
mycroft holmes
272 Points
9 years ago
Consider the fraction a/b. If it is not already in reduced form then consider the equivalent fraction x/y such that gcd(|x|, y) =1 
 
Then bx + a(-y) = 0. Also, since gcd (|x|, y) =1,  there exist integers m, n such m|x|+ny =1
 
If x>0, then mx – n(-y) =1 so that we can choose h=m, k=-y, r=n, s=-y
 
If x<0, then (-m)x -n (-y) =1 so that we can choose h = -m, k=-y, r=n, s=-y
 
Thus we can always find h,k,r,s satisfying the given conditions

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