Consider a graph $G$ with edges
$$s \stackrel{1}{\longrightarrow} v, v \stackrel{2}{\longrightarrow} t, s \stackrel{3}{\longrightarrow} t.$$
Even though all edge lengths are distinct positive integers, there exist two shortest paths, $s \rightarrow v \rightarrow t$, and $s \rightarrow t$; thus, option A is incorrect. $G$ doesn't contain a directed cycle, and yet, it doesn't have a unique shortest path; thus, option B is incorrect.
Now observe that two sums of distinct powers of two cannot be the same (imagine the numbers are written in binary); thus, option C is correct, and option D is incorrect.
