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Positive solution for linear Diophantine equations

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pyfgcr
#1
Nov27-12, 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.
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DonAntonio
#2
Nov27-12, 07:08 PM
P: 606
Quote Quote by pyfgcr View Post
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.

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
Mensanator
#3
Nov29-12, 12:40 AM
P: 105
Shouldn't that be (5,-5)?

DonAntonio
#4
Nov29-12, 05:40 AM
P: 606
Positive solution for linear Diophantine equations

Quote Quote by Mensanator View Post
Shouldn't that be (5,-5)?

Yes, you're right of course. Thanks.

DonAntonio
HallsofIvy
#5
Nov29-12, 08:08 AM
Math
Emeritus
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Thanks
PF Gold
P: 39,682
All solutions of the Diophantine equation ax+ by= c (assuming a, b, relatively prime) are of the form x= x0+ kb, y= y0- 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, x0, and y0.


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