Answer:

Consider,

a, b ∈ Q₀

a * b = (3ab/5)

= (3ba/5)

= b * a

a * b = b * a, for all a, b ∈ Q₀

a * (b * c) = a * (3bc/5)

= [a (3 bc/5)] /5

= 3 abc/25

(a * b) * c = (3 ab/5) * c

= [(3 ab/5) c]/ 5

= 3 abc /25

a * (b * c) = (a * b) * c, for all a, b, c ∈ Q₀

Then, * is associative on Q₀

Consider,

e be the identity element in Z with respect to *

a * e = a = e * a ∀ a ∈ Q₀

a * e = a and e * a = a, ∀ a ∈ Q₀

3ae/5 = a and 3ea/5 = a, ∀ a ∈ Q₀

e = 5/3 ∀ a ∈ Q₀ [because a is not equal to 0]

Hence, 5/3 is the identity element in Q₀ with respect to *.