jmc

Consider a point travelling across the internal diagonal, and let the internal diagonal have a length of $d$. The point enters a new unit cube in the $x,y,z$ dimensions at multiples of $\frac{d}{150},\frac{d}{324},\frac{d}{375}$ respectively. We proceed by using PIE. The point enters a new cube in the $x$ dimension $150$ times, in the $y$ dimension $324$ times and in the $z$ dimension, $375$ times. The point enters a new cube in the $x$ and $y$ dimensions whenever a multiple of $\frac{d}{150}$ equals a multiple of $\frac{d}{324}$. This occurs $\gcd (150,324)$ times. Similarly, a point enters a new cube in the $y,z$ dimensions $\gcd (324,375)$ times and a point enters a new cube in the $z,x$ dimensions $\gcd (375,150)$ times. The point enters a new cube in the $x,y$ and $z$ dimensions whenever some multiples of $\frac{d}{150},\frac{d}{324},\frac{d}{375}$ are equal. This occurs $\gcd (150,324,375)$ times. The total number of unit cubes entered is then $150+324+375-[\gcd (150,324)+\gcd (324,375)+\gcd (375,150))]+\gcd (150,324,375)=\boxed{768}$