## 42. Meeting on the Tower

Two delegations were to meet on the top floor of a tower that had several elevators, each capable of holding nine people. The first delegation ascended to the meeting place using a certain number of elevators that were fully occupied, plus one last elevator where there were still five empty seats. Then the second delegation did the same, with only three empty seats in the last elevator.

At the beginning of the meeting, each member of each delegation shook hands with every member of the other delegation, and each handshake was captured on a photograph. Knowing that the photographer used a roll of film that could hold nine frames, how many more photos could have been taken on the last roll of film after the photographer captured the last handshake?
42. Let $d_{1}$ be the number of members of the first delegation, and $d_{2}$ be the number of members of the second delegation. We know that

$$
\left.d_{1}=(\text { a number divisible by } 9)+4 ; d_{2}=\text { (a number divisible by } 9\right)+6 .
$$

The number of frames shot is

$$
\begin{aligned}
d_{1} \cdot d_{2} & =(\text { a number divisible by } 9)+(4 \cdot 6)= \\
& =(\text { a number divisible by } 9)+(18+6)= \\
& =(\text { a number divisible by } 9)+6 .
\end{aligned}
$$

Therefore, six frames were shot on the last roll of film. The photographer could have taken three more frames on it.