CWMO 2017 Problem 7

Let $n=2^{\alpha} \cdot q$ be a positive integer, where $\alpha$ is a nonnegative integer and $q$ is an odd number. Show that for any positive integer $m$ , the number of integer solutions to the equation $x_1^2+x_2^2+\cdots +x_n^2=m$ is divisible by $2^{\alpha +1}$ .