ZA-SAMO 2004 Problem 6

The numbers $a_1,a_2$ and $a_3$ are distinct positive integers, such that

(i) $a_1$ is a divisor of $a_2+a_3+a_2a_3$ ;

(ii) $a_2$ is a divisor of $a_3+a_1+a_3a_1$ ;

(iii) $a_3$ is a divisor of $a_1+a_2+a_1a_2$ .

Prove that $a_1,a_2$ and $a_3$ cannot all be prime.