Problem 4. Marco needs to distribute 199 apples into 60 packages. In some packages, there are $x$ apples, and in the rest, there are 3 apples. Determine all possible values of $x$.
Solution. Let there be $n$ packages, each containing 3 apples. Then the number of packages with $x$ apples would be $60-n$. From the condition of the problem, we form the equation

$$
3 n+x \cdot(60-n)=199
$$

which is equivalent to the equation $3 n-180+x \cdot(60-n)=19$ or the equation

$$
(60-n) \cdot(x-3)=19
$$

Since $x$ and $n(n \leq 60)$ are natural numbers, we get that there are two solutions, namely

$$
\left\{\begin{array}{l}
60-n=1 \\
x-3=19
\end{array}\right.
$$

from which we get $x=22$ or

$$
\left\{\begin{array}{l}
60-n=19 \\
x-3=1
\end{array}\right.
$$

from which we have that $x=4$.

## IX Department