If $n = pq$, where $p, q$ are distinct primes, it is clear that we can not arrange $p, q$ and $pq$ without $p$ and $q$ being neighbors, therefore $pq$ is not nice.

If $n$ is not a product of two distinct primes, then $n = p_1^{\alpha_1} p_2^{\alpha_2} \dots p_k^{\alpha_k}$, where $k \ge 1$ and $\alpha_i \ge 1$. A convenient way of arranging the factors of $n$, larger than $1$, on a circle is the following: we write the numbers in a succession of the form:

$n, S_1, S_2, \dots, S_{k-1}, S_k$

where:

* $S_1$ is a sequence of numbers that contains all the factors of $n$, other than $n$, that are multiples of $p_1$, the last one in the sequence being $p_1p_2$;
* $S_2$ is a sequence of numbers that contains all the factors of $n$ that are multiples of $p_2$, but not of $p_1$, the last number in the sequence being $p_2p_3$;
* $S_3$ is a sequence of numbers that contains all the factors of $n$ that are multiples of $p_3$, but are multiples of neither $p_1$ nor $p_2$, the last number in the sequence being $p_3p_4$;
... and so on.

The set $\{1, 2, \ldots, 100\}$ contains $74$ composite numbers, $30$ of which being of the form $pq$, with $p, q$ distinct primes. This leaves $44$ nice numbers.