Indeterminate System of Linear Equations

Algebra Level 4

Consider the system of equations

\[\begin{equation} \label{eq1} \begin{split} y & = 2{x}_{1}+{x}_{2} \\ y & = 3{x}_{2}+{x}_{3} \\ y & = 4{x}_{3}+{x}_{4} \\ y & = 5{x}_{4}+{x}_{5} \\ y & = 6{x}_{5}+{x}_{1}. \end{split} \end{equation}\]

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

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

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