OME 1974 Problem 3

We will designate by $Z_{(5)}$ a certain subset of the set $Q$ of the rational numbers . A rational belongs to $Z_{(5)}$ if and only if there exist equal fraction to this rational such that $5$ is not a divisor of its denominator. (For example, the rational number $13/10$ does not belong to $Z_{(5)}$ , since the denominator of all fractions equal to $13/10$ is a multiple of $5$ . On the other hand, the rational $75/10$ belongs to $Z_{(5)}$ since that $75/10 = 15/12$ ). Reasonably answer the following questions: a) What algebraic structure (semigroup, group, etc.) does $Z_{(5)}$ have with respect to the sum? b) And regarding the product? c) Is $Z_{(5)}$ a subring of $Q$ ? d) Is $Z_{(5)}$ a vector space?