Indeterminate System of Linear Equations

Algebra Level 3

Consider the system of equations

y=2x1+x2y=3x2+x3y=4x3+x4y=5x4+x5y=6x5+x1. \begin{aligned} y & = 2{x}_{1}+{x}_{2} \\ y & = 3{x}_{2}+{x}_{3} \\ y & = 4{x}_{3}+{x}_{4} \\ y & = 5{x}_{4}+{x}_{5} \\ y & = 6{x}_{5}+{x}_{1}. \end{aligned}

If all of the variables are integers, what is the minimum positive integer value of

(i=15xi)y?\left(\sum_{i=1}^{5}{x}_{i}\right) - y ?

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