## Task 3 - 240813

We are looking for a decomposition of the number 500 into four summands, with the following conditions required:

(1) All four summands are natural numbers.

(2) If you add 4 to the first summand, the result is the same as if you subtract 4 from the second summand. The same result also occurs if you multiply the third summand by 4, and also if you divide the fourth summand by 4.

Investigate whether there is only one such decomposition! If this is the case, determine it and confirm that it has the properties (1), (2)!
If a decomposition satisfies conditions (1) and (2), and if $x$ is the result mentioned four times in (2), then

$$
x-4, \quad x+4, \quad \frac{x}{4}, \quad 4 x
$$

are the four summands, so it follows that

$$
x-4+x+4+\frac{x}{4}+4 x=500 \quad \Rightarrow \quad x=80
$$

consequently, the only decomposition that can have the required properties is into the summands 76, 84, 20, 320. This decomposition has these properties; because the summands are natural numbers and satisfy the verification.