Question 165
$n$ boxes initially contain $1, 2, \ldots, n$ marbles respectively ($n \geq 1$). Charlotte first adds a marble to each box. Then she adds a marble to each box in which the number of marbles is divisible by $2$, then a marble to each box in which the number of marbles is divisible by $3$, and so on, until she adds a marble to each box in which the number of marbles is divisible by $n$. For which values of $n$ does the procedure end with exactly $n+1$ marbles in every box?