Indeterminate System of Linear Equations

Algebra Level 3

Consider the system of equations

$\begin{aligned} y & = 2{x}_{1}+{x}_{2} \\ y & = 3{x}_{2}+{x}_{3} \\ y & = 4{x}_{3}+{x}_{4} \\ y & = 5{x}_{4}+{x}_{5} \\ y & = 6{x}_{5}+{x}_{1}. \end{aligned}$

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

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