If the origin is the centroid of a triangle

Given triangle $A B C$ having vertices $A(a, 1,3), B(-2, b,-5)$ and $C(4,7, c)$ and origin is the centroid.

For a triangle the coordinates of the centroid is given by the average of the coordinates of its vertices.

Therefore,

$\Rightarrow(0,0,0)=\left(\frac{a+(-2)+4}{3}, \frac{1+b+7}{3}, \frac{3+(-5)+c}{3}\right)$

Now by comparing the each point we get

$\Rightarrow \frac{a+2}{3}=0, \therefore a=-2$

$\Rightarrow \frac{\mathrm{b}+8}{3}=0, \therefore \mathrm{b}=-8$

$\Rightarrow \frac{\mathrm{c}-2}{3}=0, \therefore \mathrm{c}=2$

$\Rightarrow \frac{c-2}{3}=0, \therefore c=2$