10.2.1. (12 points) Find the greatest integer value of $a$ for which the equation

$$
\sqrt[3]{x^{2}-(a+7) x+7 a}+\sqrt[3]{3}=0
$$

has at least one integer root.
Answer: 11.

Solution. For the desired values of $a$, the numbers $x-a$ and $x-7$ are integers, and their product is -3, so there are 4 possible cases:

$$
\left\{\begin{array} { l } 
{ x - a = 1 , } \\
{ x - 7 = - 3 ; }
\end{array} \quad \left\{\begin{array} { l } 
{ x - a = 3 , } \\
{ x - 7 = - 1 ; }
\end{array} \quad \left\{\begin{array} { l } 
{ x - a = - 1 , } \\
{ x - 7 = 3 ; }
\end{array} \quad \left\{\begin{array}{l}
x-a=-3 \\
x-7=1
\end{array}\right.\right.\right.\right.
$$

from which we find respectively

$$
\left\{\begin{array} { l } 
{ a = 3 , } \\
{ x = 4 ; }
\end{array} \quad \left\{\begin{array} { l } 
{ a = 3 , } \\
{ x = 6 ; }
\end{array} \quad \left\{\begin{array} { l } 
{ a = 1 1 , } \\
{ x = 1 0 ; }
\end{array} \quad \left\{\begin{array}{l}
a=11 \\
x=8
\end{array}\right.\right.\right.\right.
$$

Thus, the desired values are $a=3$ and $a=11$, of which the largest is 11.