2. The fraction $A=\frac{2 x-14}{x-4}$, where $x$ is an integer different from 4.

a) For which values of the number $x$ is the fraction $A$ an integer?

b) For which values of the natural number $x(x>4)$ does the fraction $A$ have the smallest value?
Solution. a) We have $A=\frac{2 x-14}{x-4}=\frac{2(x-4)-6}{x-4}=2-\frac{6}{x-4}$. This expression can be an integer if and only if 6 is divisible by $x-4$, which is possible if and only if $x-4= \pm 1, \pm 2, \pm 3, \pm 6$, i.e., $x \in\{-2,1,2,3,5,6,7,10\}$.

b) The fraction $A$ will have the smallest value if the fraction $\frac{6}{x-4}$ has the largest value, which is satisfied for $x-4=1$, i.e., $x=5$.