6. (10 points) As shown in the figure, a rectangular block with dimensions $15 \mathrm{~cm}, 5 \mathrm{~cm}, 4 \mathrm{~cm}$ has a smaller rectangular block with dimensions $y \mathrm{~cm}, 5 \mathrm{~cm}, x \mathrm{~cm}$ cut out from it $\left(x, y\right.$ are integers). The remaining volume is $120 \mathrm{~cm}^{3}$. Therefore, $x+y=$ . $\qquad$
【Analysis】The volume of the cut-off rectangular prism $=$ the volume of the larger rectangular prism - the volume of the remaining part. Based on this, we list the expression $15 \times 5 \times 4-120$ to find the volume of the cut-off rectangular prism. Then, given that $x, y$ are integers, we can solve the problem using the properties of integers.

【Solution】
$$
\begin{array}{l}
15 \times 5 \times 4-120 \\
=300-120 \\
=180\left(\mathrm{~cm}^{3}\right)
\end{array}
$$

Thus, $5 x y=180$, which means $x y=36$.
Since $x, y$ are integers, and $0<x<4, 0<y<15$,
$x$ is $3 \mathrm{~cm}, y$ is $12 \mathrm{~cm}, x+y=15 \mathrm{~cm}$.
Therefore, the answer is: 15.