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Grade: 9
        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
7 years ago

Answers : (2)

Arun Kumar
IIT Delhi
askIITians Faculty
256 Points
							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
5 years ago
mycroft holmes
272 Points
							
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
5 years ago
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