Question 23
Kärt writes the fractions $\frac{1}{2}$ and $\frac{1}{3}$ on the blackboard and Märt writes 10 positive integers on the paper, which he does not show to Kärt. Then Kärt starts to write fractions on the blackboard by the following rule: on each step she chooses two fractions $\frac{a}{b}$ and $\frac{c}{d}$ which are already on the blackboard and writes on the blackboard the fraction $\frac{a+c}{b+d}$ after reducing. Can Kärt always choose the fractions so that after a number of steps she writes on the blackboard a fraction whose denominator is coprime with all the numbers Märt has written on the paper?