jmc

Note that, modulo 6, $35\equiv -1$ and $16\equiv 4$. Therefore, $35^{1723}-16^{1723}\equiv (-1{)}^{1723}-4^{1723}\equiv -1-4^{1723}$. Note that $4^2\equiv 4$ modulo 6, so $-1-4^{1723}\equiv -1-4\equiv -5\equiv \boxed{1}$ modulo 6.