Question 21
Distinct positive integers $a$, $b$, $c$, $d$ satisfy
$$
\begin{cases}
a|b^2 + c^2 + d^2 \\
b|a^2 + c^2 + d^2 \\
c|a^2 + b^2 + d^2 \\
d|a^2 + b^2 + c^2
\end{cases} .
$$
Also it is known that none of them is larger than the product of three others. What is the largest possible number of primes among them?
$$
\begin{cases}
a|b^2 + c^2 + d^2 \\
b|a^2 + c^2 + d^2 \\
c|a^2 + b^2 + d^2 \\
d|a^2 + b^2 + c^2
\end{cases} .
$$
Also it is known that none of them is larger than the product of three others. What is the largest possible number of primes among them?