If the plane $7x+11y+13z=3003$ meets the coordinate axes at A,B,C then the centroid of the $∆ABC$ is $\left(a,b,c\right)$ then the value of $a+b+c$ is  

Equaiton of the given plane is $7x+11y+13z=3003$

The interept form of the plane is 

                 $$\begin{gathered} \frac{7x}{3003}+\frac{11y}{3003}+\frac{13z}{3003}=1 \\ \Rightarrow \frac{x}{429}+\frac{y}{273}+\frac{z}{231}=1 \end{gathered}$$

The plane $7x+11y+13z=3003$ meets the coordinate axes in $A\left(429,0,0\right),B\left(0,273,0\right),C\left(0,0,231\right)$

The centroid of the triganle $ABC$ is $\left(a,b,c\right)=\left(\frac{429}{3},\frac{273}{3},\frac{231}{3}\right)=\left(143,91,77\right)$

Therefore, $a+b+c=143+91+77=\boxed{311}$