Evdokimov M.A.

Sages \$ will be given three cards each, and each will see only their own cards. After the distribution, the sages spoke in turn as follows:

\$AS: "One of my cards has the number 8".

\$B\$: "All the numbers on my cards are prime".

\$C\$: "All the numbers on my cards are composite, and they have a common prime divisor".

\$D\$: "Then I know what cards each of you has".

What cards does \$ A have?
Prime numbers can only be at $\$ \mathrm{~A}, \mathrm{~B} \$$ and $\$ \mathrm{D} \$$$. On the cards $\$ \mathrm{~B} \$$ - three out of five possible prime numbers $(2,3,5,7,11)$. The other two prime numbers are with $\$ D \$$, otherwise he would not know which of the prime numbers are with $\$ \mathrm{~B} \$$, and which are with $\$ \mathrm{~A} \$$.

On the cards \$C\$ there can be triples $(4,6,10),(4,6,12),(4,10,12),(6,10,12)$ or $(6,9,12)$. Only if $\$ D \$$ has 6 or 12, he can determine which exact triple is with $\$ C \$$.

So, $\$ D \$$ has two prime numbers and one of the numbers 6 and 12, $\$ C \$$ respectively - $4,6,10$, or $4,10,12$, $\$ B \$-$ three prime numbers, and $\$$ A $\$$ - numbers $1,8,9$.

## Otvet

1,8 and 9.