## Task 1 - 200831

Uwe tells:

"During the winter break, we went on a trip to the Harz with a group of travelers. No more than 80 people participated, and there were exactly 3 fewer men than women and exactly 20 more adults than children. On the way, we were divided into exactly 7 groups of equal number of people."

Determine all possibilities to specify the numbers of men, women, and children so that Uwe's statements are true!
If Uwe's statements apply to a specification of the number of men, women, and children, and $m$ is the number of participating men, then:

The number of participating women is $m+3$, so the number of adults is $2m+3$; it follows that $2m+3 \geq 20$ (1)

and the number of children is $2m+3-20$.

Thus, $m+m+3+2m+3-20=4m-14$ is the total number of participants. Therefore, on the one hand, $4m-14 \leq 80 \quad(2)$

on the other hand, $4m-14$ is divisible by 7. This also applies to $4m$; thus, due to the coprimality of 4 and 7, $m$ is divisible by 7.

If $m \leq 7$, then $2m+3 \leq 17$, contradicting (1); if $m \geq 28$, then $4m-14 \geq 98$, contradicting (2).

Therefore, (1), (2), (3) can only be satisfied if $m=14$ or $m=21$. Thus, only the numbers of people given in the following table can match Uwe's statements. It is also evident from the table that they indeed match these statements:

| Men | Women | Adults | Children | Participants |
| :---: | :---: | :---: | :---: | :---: |
| $m$ | $m+3$ | $2m+3$ | $2m-17$ | $4m-14$ |
| 14 | 17 | 31 | 11 | 42 |
| 21 | 24 | 45 | 25 | 70 |