4) A flea is on the number 12 of a clock face. It chooses a natural number $n$ between 1 and 12, inclusive, and starts jumping $n$ numbers on the clock face, in a clockwise direction (for example, if $n=3$, after the first jump it is on 3, after the second it is on 6, and so on). After 12 jumps, for the first time, it finds itself back on the number 12 of the clock face. In how many distinct ways could it have chosen $n$?
(B) 2,
(C) 4,
(D) 6,
(E) 12.
4. The answer is $(\mathbf{C})$.

Let's start by observing that the flea returns to 12 after a number $s$ of jumps if and only if $s \cdot n$ is a multiple of 12. If $n$ and 12 had a common divisor $d$, then in $\frac{12}{d}<12$ jumps the flea would travel $n \frac{12}{d}=12 \frac{n}{d}$ hours, and thus would return to 12. Since this cannot happen, $n$ and 12 must be coprime. On the other hand, if $n$ and 12 are coprime, the smallest multiple of $n$ that is also a multiple of 12 is $12 n$ (the least common multiple of $n$ and 12), so the flea does not return to 12 before 12 jumps. We conclude that the flea could have chosen $n$ if and only if $n$ and 12 are coprime, i.e., the acceptable $n$ are the following four numbers: 1, 5, 7, 11. [Problem proposed by K. Kuzmin.]