How many solutions does the following equation have:
$x_1+2x_2+3x_3+4x_4+5x_5+6x_6+7x_7+8x_8+9x_9+10x_{10}\equiv0\mod11$
where
$x_{1...9} \in \{0,1,2,3,4\ ...\ 8,9\}$ and $x_{10}\in\{0,1,2,3,4\ ...\ 9,10\}$?
I've being trying to solve the equation with generating functions. The closed form that I've arrived at is:
$(1-x^{10})(1-x^{20})(1-x^{30})...(1-x^{110}) \above 1pt (1-x)(1-x^2)(1-x^3)...(1-x^{10})$
where the sum of coefficients of all the terms of the form $x^n$ where $n\equiv0\mod11$, is the answer to the original problem. But it seems to make it even harder.
Can somebody help me with the solution or just give a hint of what else I might try? I swear I've spent with the problem quite a while now, so I think I simply need someone to have a fresh look on it.