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Positive solution for linear Diophantine equations 
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#1
Nov2712, 06:28 PM

P: 22

The linear Diophantine equations: ax+by=c, a,b,c is natural numbers.
If c is a multiple of gcd(a,b), there is infinite integer solutions, and I know how to find x,y. However, I wonder how to find positive integer solution x,y only. 


#2
Nov2712, 07:08 PM

P: 606

They may not exists. For example, the equation [itex]\,7x+6y=5\,[/itex] cannot have positive solutions, but it has solutions, like [itex]\,(5,6)\,[/itex] DonAntonio 


#3
Nov2912, 12:40 AM

P: 105

Shouldn't that be (5,5)?



#4
Nov2912, 05:40 AM

P: 606

Positive solution for linear Diophantine equations
Yes, you're right of course. Thanks. DonAntonio 


#5
Nov2912, 08:08 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,344

All solutions of the Diophantine equation ax+ by= c (assuming a, b, relatively prime) are of the form x= x_{0}+ kb, y= y_{0} ka for k any integer. If you want both x and y positive, you must be able to choose k so that those are postive. Whether that is possible, of course, depends on a, b, x_{0}, and y_{0}.



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