MathDB

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?

Problem Statement