We have the equation $$x_1^2 + \dots + x_{36}^2 = 49$$Suppose $x_1 = 4$, then $$x_2^2 + \dots + x_{36}^2 = 33$$which is not possible, so $x_i \le 3$ for all $i$. Also note there is at most one $3$. If there is no $3$, then modulo $3$ gives the result. If there is one $3$, modulo $3$ still gives the result.