1.10. (CSSR, 79). Find all natural numbers $n>2$, not exceeding the number 10000000 and possessing the following property: any number $m$, coprime with $n$ and satisfying the inequalities $1<m<n$, is prime.
1.10. Let the number $n$ have the property required by the problem. If it is not divisible by 2, then $np_{k+1}^{2}
$$

which contradicts the necessary condition

$$
p_{1} p_{2} . . p_{k} \leqslant n < p_{k+1}^{2}
$$

The obtained contradiction shows that the considered case is impossible. Thus, it is proven that all possible values of $n$ belong to the set

$$
\{3 ; 4 ; 6 ; 8 ; 12 ; 18 ; 24 ; 30\}
$$

Verification shows that each number in this set has the property specified in the problem condition.