Fix a positive integer $r$. Describe the solutions to the system of equations given by:

$$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$

Example: In the case that $r=3$, $(X_1,X_2,X_3)=(1,5,13)$ is a solution, because $1^2+5^2+13^2=195=3\cdot 5\cdot 13$ is divisible by $1$, $5$ and $13$.

Motivation: The degrees of the irreducible characters $\text{Irr}(G)$ of a finite group $G$ satisfy the following relation:

$$\begin{equation}\sum_{\chi\in\text{Irr}(G)}\chi(1)^2=|G|\end{equation}$$

Furthermore, for every $\chi\in\text{Irr}(G)$, $\chi(1)$ divides $|G|$. So the broader question is: is there some way to arithmetically characterize sequences of positive integers which are the irreducible degrees of some finite group?

The example I gave above does not qualify. No group has irreducible degrees $(1,5,13)$. Such a group would have $195$ elements, but since $195$ is a squarefree number, our hypothetical group with $195$ elements would have to be solvable. But nontrivial solvable groups must have nonprincipal linear characters, so $(1,5,13)$ cannot be its set of irreducible degrees.