HMMT November 2021 T2

Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a+b=c$ or $a \cdot b=c$ for some variables or integers $a, b, c$. Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:

$$\begin{array}{ll} x & z=15 \\ x & y=12 \\ x & x=36 \end{array}$$

If $x, y, z$ are integer solutions to the original system, find the sum of all possible values of $100 x+10 y+z$. Proposed by: David Vulakh