## Task 1 - 200731

A natural number $z$ is required to be decomposable into four summands that satisfy the following conditions:

The first summand is two thirds of the number $z$, the second summand is one quarter of the first summand, the third summand is four fifths of the second summand, the fourth summand is one quarter of the third summand, and the third summand is 48.

Investigate whether these conditions can be met!

If this is the case, determine all natural numbers $z$ and their decompositions into four summands that satisfy these conditions!
When a natural number $z$ is decomposed into four summands $s_{1}, s_{2}, s_{3}, s_{4}$ such that the given conditions are satisfied, then

$$
\begin{array}{rlrl}
z=s_{1}+s_{2}+s_{3}+s_{4} & \text { (1) } & s_{1} & =\frac{2}{3} z \\
s_{3}=\frac{4}{5} s_{2}
\end{array}
$$

$$
\begin{array}{r}
s_{2}=\frac{1}{4} s_{1} \\
s_{3}=48
\end{array}
$$

From (5) and (6) it follows (7) $s_{4}=\frac{1}{4} \cdot 48=12$.

From (6) and (4) it follows (8) $48=\frac{4}{5} \cdot s_{2}$, thus $s_{2}=\frac{5}{4} \cdot 48=60$.

From (8) and (3) it follows (9) $60=\frac{1}{4} \cdot s_{1}$, thus $s_{1}=4 \cdot 60=240$.

From (1) and (6) to (9) it follows $z=240+60+48+12=360$.

Therefore, only the number $z=360$ and its decomposition into $s_{1}=240, s_{2}=60, s_{3}=48, s_{4}=12$ can satisfy the conditions of the problem statement.