Exercise 10. Using the numbers from 1 to 22 exactly once each, Antoine writes 11 fractions: for example, he can write the fractions $\frac{10}{2}, \frac{4}{3}, \frac{15}{5}, \frac{7}{6}, \frac{8}{9}, \frac{11}{19}, \frac{12}{14}, \frac{13}{17}, \frac{22}{21}, \frac{18}{16}$, and $\frac{20}{1}$.

Antoine wishes to have as many fractions with integer values as possible among the fractions written: in the previous example, he wrote three fractions with integer values: $\frac{10}{2}=5, \frac{15}{5}=3$, and $\frac{20}{1}=20$. What is the maximum number of fractions that can have integer values?
Solution to Exercise 10 First, it is possible to form 10 whole numbers with the fractions

$$
\left\{\frac{22}{11}, \frac{20}{10}, \frac{18}{9}, \frac{16}{8}, \frac{14}{7}, \frac{12}{6}, \frac{21}{3}, \frac{19}{1}, \frac{15}{5}, \frac{4}{2}, \frac{17}{13}\right\}
$$

Furthermore, it is impossible to have 11 whole fractions. Indeed, there are 3 prime numbers that have no smaller multiples than 22: 19, 17, and 13. Thus, the only way for any of these three numbers to form a whole fraction is with the number 1, which can only be used with one of these three numbers.

Graders' Comments: The exercise was generally very well done. However, some students, although noting the importance of 13, 17, and 19, did not think to combine two of them in the same fraction, and therefore only obtained 9 whole fractions, which is a pity. Other students were content with a configuration of 10 fractions without justifying that it is optimal. Finally, we remind you that $\frac{19}{8}$ is definitely not an integer.