João's father has between 200 and 300 books in his library. One fifth of these books are in English, one seventh in French, one quarter in Italian, and the rest are in Spanish. What is the total number of Spanish books in this library?

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Solution

The number of books is a multiple of 5, 7, and 4. Since these numbers do not have common prime factors, it must necessarily be a multiple of \(5 \cdot 7 \cdot 4 = 140\). In the given range, there is a unique multiple of 140, namely the number 280. Therefore, the number of books that are not in Spanish is

$$
\frac{280}{5}+\frac{280}{7}+\frac{280}{4}=56+40+70=166
$$

Thus, there are \(280-166=114\) Spanish books.