MAT3702 Assignment 2 2026 (Answer Guide) – Due 19 August 2026

MAT3702 Assignment 2 2026 (Answer Guide) – Due 19
August 2026

ASSIGNMENT 2
Question 1
Prove that b ∣ a if and only if (−b) ∣ a.

Proof:
Recall the definition of divisibility:

For integers a, b, we say b ∣ a if there exists k ∈ Z such that

a = bk.



(⇒) Suppose b ∣ a
Then there exists k ∈ Z such that

a = bk.

But we can write:

a = (−b)(−k).

Since −k ∈ Z, it follows that

(−b) ∣ a.

,(⇐) Suppose (−b) ∣ a
Then there exists m ∈ Z such that

a = (−b)m.

But

a = b(−m),

and since −m ∈ Z, we conclude that

b ∣ a.



Since both implications hold,

b ∣ a ⟺ (−b) ∣ a.

□




Question 2
If a ∣ b and b ∣ c, prove that a ∣ c.

Proof:
Assume:

a ∣ b and b ∣ c.

By definition of divisibility:

There exists k ∈ Z such that

b = ak,

and there exists m ∈ Z such that

c = bm.

Substitute b = ak into the second equation:

c = (ak)m.

, Thus,
c = a(km).


Since km ∈ Z, it follows that
a ∣ c.


Therefore divisibility is transitive.

□




Question 3
Let R, S be rings and consider the subsets of R × S:

R = {(r, 0S) ∣ r ∈ R}

S = {(0R, s) ∣ s ∈ S}

where 0R, 0S are the zero elements in R and S.



(i) If R = Z3 and S = Z5, find R and S.
Recall:

Z3 = {0, 1, 2}

Z5 = {0, 1, 2, 3, 4}

Thus:

R = {(0, 0), (1, 0), (2, 0)}

S = {(0, 0), (0, 1), (0, 2), (0, 3), (0, 4)}



(ii) Show that R is a subring of R × S
We verify the subring test.