## Task 4 - 090614

A working group received a sum of exactly $240 \mathrm{M}$ as an award for very good performance.

If this money had been distributed evenly among all members of the working group, each member would have received an integer amount (in Marks). However, the members decided to spend the $240 \mathrm{M}$ together on a hiking trip.

Exactly three of the members could not participate in the hiking trip, as a result, with the money distributed evenly among all participants of the hiking trip, each participant had exactly $4 \mathrm{M}$ more available than if the money had been distributed evenly among all members.

Determine the number of members in the working group!
Let the number of students in the working group be $x$. With an even distribution of the money, each student thus receives $a=\frac{240}{x}$, which means that $x$ must be a divisor of 240.

If 3 members do not participate, the share for the remaining members increases to $a+4$, i.e., $a+4=\frac{240}{x-3}$, which means that $x-3$ must also be a divisor of 240.

The divisors $x$ of 240 for which $x-3>0$ are

$$
4,5,6,8,10,12,15,16,20,24,30,40,48,60,80,120,240
$$

Only the pairs $(8 ; 5)$ and $(15 ; 12)$ are pairs of the form $(x ; x-3)$. Substituting the first equation into the second gives: $\frac{240}{x}+4=\frac{240}{x-3}$.

Substituting the two pairs, it turns out that only $x=15, x-3=12$ satisfy the last equation. Therefore, the working group had 15 members.