MAT3702 Assignment 2 Memo | Due 19 August 2026

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, PLEASE USE THIS DOCUMENT AS A GUIDE TO ANSWER YOUR ASSIGNMENT


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1. Prove that b a if and only if {- b) Ia
To prove an "if and only if" statement, we must prove the implication in both directions.

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Direction 1: If b a, then (-b) I a
1. By definition, if b I a, then there exists an integer k such that:

a= bk

2. We can rewrite this equation to involve -b by multiplying by (-1) (-1):

a= (-b) (-k)

3. Since k is an integer, -k is also an integer.

4. Therefore, by the definition of divisibility, ( -b) I a.


Direct ion 2: If (-b) I a, then b I a
1. By definition, if (-b) I a, then there exists an integer m such that:


a = (-b)m

2. This can be rewritten as:

a= b(-rn)

3. Since mis an integer, -m is also an integer.
4. Therefore, b I a.

Conclusion: Since both directions hold, b I a <⇒ (- b) I a.