Generating set of Toric ideal in Integer Programming

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I’m stuck with one question to find the generating set of toric ideal.

Here is how the integer programming question is formatted:

min 5x+7y+z
s.t. x+6y+8z=24, x,y,z /in N

Normally I know we can relax to LP problem and do branch and bound, anyone has any direction on how to solve this in an abstract algebra way?