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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

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8 years ago Arun Kumar
IIT Delhi
256 Points
```							Hibut 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
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6 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
```
6 years ago
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