8.4. Let's call the average magic of a set of numbers the ratio of their sum to their product. Initially, several (more than one) different natural numbers are written on the board. After the smallest of them is erased, the average magic of all the numbers written on the board increases threefold. Find what the numbers on the board were.
Answer: 4 and $12 ; 4,5,7$.

Solution 1: Let the original sum be $s+a$, the product be $ap$, and the smallest number be $a$. Then we get and simplify the following relationship:

$$
\begin{gathered}
3(s+a) / a p=s / p \\
3 s / a+3=s \\
1 / a+1 / s=1 / 3
\end{gathered}
$$

On one hand, this expression can be estimated as follows:

$$
\begin{gathered}
2 / a>1 / a+1 / s=1 / 3 \\
6>a
\end{gathered}
$$

On the other hand:

$$
\begin{aligned}
1 / a & 12$ ).

If there are three numbers and the second is greater than 5, i.e., at least 6, then the third is also greater than 6, but then $s>12$, so if there were three numbers, the first must be 4, the second 5, and the remaining one is uniquely determined.

Criteria: Only the answer, answer with verification - 1 point. Obtaining the inequality 6 $>a-$ plus 1 point. Obtaining the inequality $30$. Dividing the first equation by the second and denoting $N=a_{2}+\cdots+a_{n}$. We get $\frac{a_{1}+N}{N}=\frac{a_{1}}{3}$, or $a_{1}=\frac{3 N}{N-3}$.

GCD $(3 N, N-3)=$ GCD $(9, N-3)$ and equals $N-3$, since $3 N$ is divisible by $N-3$ exactly. From here, since GCD $(9, N-3)=1,3$ or 9, we get three cases: $N-3=1,3$ or 9.

1) $N=4$, hence $a_{1}=12$. But then the sum of all numbers $a_{1}+N=16$, which means $a_{1}$ could not have been the smallest number on the board.
2) $N=6$, hence $a_{1}=6$. Similarly, we get a contradiction.
3) $N=12$, hence $a_{1}=4$. Then we get that the sum of several different natural numbers, greater than 4, is 12. We get two cases: either $a_{2}=12$, or $a_{2}=5, a_{3}=7$. It is easy to verify that both of these cases fit.

Criteria: only the answer, answer with verification - 1 point. Obtaining that GCD(9, $N-3)=1,3$ or 9 - plus three points. Both cases 1) and 2) are analyzed - plus 1 point.