jmc

The amount of powdered sugar on a given donut hole is given by the surface area of the donut hole. The surface area of a sphere with radius $r$ is $4\pi r^2$, so Niraek's donut holes each have surface area $4\pi \cdot 6^2=144\pi$ square millimeters. Similarly, Theo's donut holes each have surface area $4\pi \cdot 8^2=256\pi$ square millimeters and Akshaj's donut holes each have surface area $4\pi \cdot 10^2=400\pi$ square millimeters.

To determine the amount of powdered sugar used the first time all three workers finish at the same time, we compute the lowest common multiple of $144\pi$, $256\pi$, and $400\pi$. $144=2^4\cdot 3^2$, $256=2^8$, and $400=2^4\cdot 5^2$, so the desired LCM is $2^8\cdot 3^2\cdot 5^2\pi$. The number of donut holes Niraek will have covered by this point is $\frac{2^8\cdot 3^2\cdot 5^2\pi }{144\pi }=2^4\cdot 5^2=\boxed{400}$.