MAT3702
Assignment 2
Due 19 August 2026
, Question 1
Problem Statement
Prove that 𝑏 ∣ 𝑎 if and only if (−𝑏) ∣ 𝑎.
Solution
Assume 𝑏 ∣ 𝑎 .
Then there exists an integer 𝑘 such that
𝑎 = 𝑏𝑘.
Multiply both sides by −1:
𝑎 = (−𝑏)(−𝑘).
Since −𝑘 is an integer, this shows that (−𝑏) ∣ 𝑎.
Now assume (−𝑏) ∣ 𝑎.
Then there exists an integer 𝑚 such that
𝑎 = (−𝑏)𝑚.
Rewrite this as
𝑎 = 𝑏 (−𝑚).
Since −𝑚 is an integer, 𝑏 ∣ 𝑎 .
Final Answer
𝑏 ∣ 𝑎 ⇔ (−𝑏) ∣ 𝑎.
Assignment 2
Due 19 August 2026
, Question 1
Problem Statement
Prove that 𝑏 ∣ 𝑎 if and only if (−𝑏) ∣ 𝑎.
Solution
Assume 𝑏 ∣ 𝑎 .
Then there exists an integer 𝑘 such that
𝑎 = 𝑏𝑘.
Multiply both sides by −1:
𝑎 = (−𝑏)(−𝑘).
Since −𝑘 is an integer, this shows that (−𝑏) ∣ 𝑎.
Now assume (−𝑏) ∣ 𝑎.
Then there exists an integer 𝑚 such that
𝑎 = (−𝑏)𝑚.
Rewrite this as
𝑎 = 𝑏 (−𝑚).
Since −𝑚 is an integer, 𝑏 ∣ 𝑎 .
Final Answer
𝑏 ∣ 𝑎 ⇔ (−𝑏) ∣ 𝑎.
