Question 10
There exists a unique positive integer $a$ for which the sum $$
U = \sum_{n=1}^{2023} \left\lfloor \frac{n^2 - na}{5} \right\rfloor
$$ is an integer strictly between $-1000$ and $1000.$ For that unique $a,$ find $a + U.$
(Note that $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x.$)
U = \sum_{n=1}^{2023} \left\lfloor \frac{n^2 - na}{5} \right\rfloor
$$ is an integer strictly between $-1000$ and $1000.$ For that unique $a,$ find $a + U.$
(Note that $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x.$)