Question 108
Find all the positive integer $N$ such that there exists $M$ satisfying the following two conditions:
a. $M$'s first few digits coincide with $N$.
b. Let $S$ be a number obtained from $M$ by transferring the digits that express $N$ from the beginning of $M$ to the end of $M$. Then $S \cdot N = M$.
(For example, $M = 460100021743857360295716$ satisfies the conditions, for $N = 46$).
a. $M$'s first few digits coincide with $N$.
b. Let $S$ be a number obtained from $M$ by transferring the digits that express $N$ from the beginning of $M$ to the end of $M$. Then $S \cdot N = M$.
(For example, $M = 460100021743857360295716$ satisfies the conditions, for $N = 46$).