Nines Divisibility

Solution 1: By the rules of divisibility, $n + 8 + 7 + 8 + n$ must be a multiple of $9$ . Thus, $2n +23 \equiv 0 \pmod{9}$ , which gives $2n \equiv -14 \pmod{9} \Rightarrow n \equiv -7 \equiv 2 \pmod{9}$ . Since $n$ is a digit, $n= 2$ .

Solution 2: We can check that $2+8+7+8+2=27$ is a multiple of $9$ , hence by the rules of divisibility, $27872$ is a multiple of $9$ . It remains to check that no other digit $n$ will work.