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Given a natural number q ? 1, define a relation ? on the set Z by...

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Advance Math

Given a natural number q ? 1, define a relation ? on the set Zby x ? y
if x - y is divisible by q.
(i) Show that ? is an equivalence relation.
We will denote the set of equivalence classes defined by ? withZ=qZ. Also
let x mod q denote the equivalence class to which an integer xbelongs.
(ii) Check that the operations

(x
(x		mod q) + (y
mod q) · (y		mod q) = (x + y)
mod q) = (x · y)		mod q;
mod q;

are well-defined on Z=qZ.
(iii) With the operations as defined above, show that Z=qZ is not afield
if q is not prime.

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