The product of three natural numbers is 600. If we decreased one of the factors by 10, the product would decrease by 400. If we increased that one factor by 5 instead, the product would double. Which three natural numbers have this property?

(L. Hozová)

Hint. From each declarative sentence in the problem, one factor can be directly determined.
Let's first work with the second sentence: by reducing one of the factors by 10, the product decreases by 400. Meanwhile, 400 is two-thirds of 600, so 10 must be two-thirds of the mentioned factor. Therefore, the number is 15.

Next, let's work with the third sentence: by increasing one of the factors by 5, the product doubles. By increasing it by 5, this factor also doubles. Therefore, the factor is 5.

From the first sentence of the problem, we know that the product of all factors is 600, and two of them are mentioned above. The third one is \(600 \div 15 \div 5 = 8\).

The numbers 5, 8, and 15 satisfy the information given in the problem.

Another hint. The problem leads to three equations with three unknowns.

Another solution. Since it is not clear from the problem whether we are reducing/increasing the same factor or a different one each time, we need to consider both possibilities. In each case, let the unknown natural numbers be \(x, y\), and \(z\).

1. Assume that the problem refers to two different factors. In this case, we can rewrite the information from the problem as follows:

\[
\begin{aligned}
x y z & = 600, \\
(x-10) y z & = 200, \\
x(y+5) z & = 1200.
\end{aligned}
\]

The second equation, after expanding, is \(x y z - 10 y z = 200\). Since \(x y z = 600\), after rearranging, we get \(10 y z = 400\), i.e., \(y z = 40\). From the equation \(x y z = 600\), it follows that \(x \cdot 40 = 600\), i.e.,

\[
x = 15.
\]

Similarly, the third equation, after expanding, is \(x y z + 5 x z = 1200\). Since \(x y z = 600\), after rearranging, we get \(5 x z = 600\), i.e., \(x z = 120\). Since we already know \(x = 15\), it must be

\[
z = 8.
\]

Substituting back into the equation \(x y z = 600\), we have \(120 y = 600\), from which it follows that

\[
y = 5.
\]

2. Assume that the problem refers to the same factor twice. In this case, we can write:

\[
\begin{aligned}
x y z & = 600, \\
(x-10) y z & = 200, \\
(x+5) y z & = 1200.
\end{aligned}
\]

Similarly as above, we expand the second and third equations, substitute \(x y z = 600\), and after rearranging, we get \(10 y z = 400\), i.e., \(y z = 40\), and \(5 y z = 600\), i.e., \(y z = 120\). Since \(40 \neq 120\), we see that the initial assumption cannot be true.

The problem refers to two different factors; the three natural numbers that satisfy the given property are 5, 8, and 15.