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I'm looking for an algorithm to create a very simple (2 equations, 2 unknowns) linear system of equations that consists purely of integers. Specifically, a way to create a system of equations of integers and knowing that it can only be solved by integer answers, without actually solving it.

a

a

where a

The only thing I can think of is using a determinant which gives

x

x

and that the numerator must be a multiple of the denominator.

What do I do now? Am I even on the right path?

a

_{11}x_{1}+a_{12}x_{2}=b_{1}a

_{21}x_{1}+a_{22}x_{2}=b_{2}where a

_{11}, a_{12}, a_{21}, a_{22}, x_{1}, x_{2}, b_{1}, b_{2}are all integers.The only thing I can think of is using a determinant which gives

x

_{1}= (a_{22}b_{1}-a_{12}b_{2}) / (a_{11}a_{22}-a_{12}a_{21})x

_{2}= (a_{11}b_{2}-a_{21}b_{1}) / (a_{11}a_{22}-a_{12}a_{21})and that the numerator must be a multiple of the denominator.

What do I do now? Am I even on the right path?

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