MlT37D2
r r r r r
lltl: QUESTIONS Fltlltll ANSWERED
t t t t t
DUE: 19 AUGUSli 2026 +
DISCLAIMER: I
- TKIJ DOCUMENflS DlrREFE~AND71 ~1'CE PUR SES-
DNL.Y. I DD NDTrTAKE RESPDNSIBIUITY 4FDR AN~ PLAGIARI M. ~MISUSS. DR
ACADEMIC MISCONDUCT RESULTING FROM THEIR USE. IT I YOUR j ~
RESPONSIBILITY Tq ENsut DRIGINAL~TY lN coMPLIIANEE WITH
t ~....:.... ........~
+-
----;RELEVA T GUIDELINES DR STANDARDS.
, PLEASE USE THIS DOCUMENT AS A GUIDE TO ANSWER YOUR ASSIGNMENT
I
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.
I
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.
r r r r r
lltl: QUESTIONS Fltlltll ANSWERED
t t t t t
DUE: 19 AUGUSli 2026 +
DISCLAIMER: I
- TKIJ DOCUMENflS DlrREFE~AND71 ~1'CE PUR SES-
DNL.Y. I DD NDTrTAKE RESPDNSIBIUITY 4FDR AN~ PLAGIARI M. ~MISUSS. DR
ACADEMIC MISCONDUCT RESULTING FROM THEIR USE. IT I YOUR j ~
RESPONSIBILITY Tq ENsut DRIGINAL~TY lN coMPLIIANEE WITH
t ~....:.... ........~
+-
----;RELEVA T GUIDELINES DR STANDARDS.
, PLEASE USE THIS DOCUMENT AS A GUIDE TO ANSWER YOUR ASSIGNMENT
I
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.
I
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.
