+0  
 
0
49
1
avatar

Let n ∈ Z.
(1) Give a direct proof of the following: If n is odd, then 4 | n
2 − 1.

(2) State the contrapositive of the implication in (1).
(3) Give a direct proof of the following: If n is even, then 4 doesn’t divide n
2 − 1.

(4) State the contrapositive of the implication in (3).
(5) State the conjunction of the implications in (1) and (3) using “if and only if”.

 Apr 21, 2020
 #1
avatar+20767 
+1

1)  Direct proof of:  If n is odd, then 4 divides (n2 - 1).

 

      If n is odd, then there exists an integer k such that  2k + 1  =  n.

      Then:  n2 - 1  =  (2k + 1)2 - 1  =  (4k2 + 4k + 1) - 1  =  4k2 + 4k  =  4(k2 + k)

                  and  4(k2 + k) / 4  =  k2 + k

       Also;  since k is an integer, k2 + k is an integer; therefore the proof is complete.

 

2)  Contrapositive of:  "if n is odd, then 4 divides (n2 - 1)" is "if 4 divides (n2 - 1), then n is odd".

 

3)  Direct proof of:  If n is even, then 4 doesn't divide n2 -1.

 

     If n is even, then there exists an integer k such that 2k = n.

     Then:  n2 - 1  =  (2k)2 - 1 =  4k2 - 1

                 and   (4k2 - 1) / 4  =  4k2/4 - 1/4  =  k2 - 1/4

      estblishing a remainder of 1/4 (or, as an integer, either -3 or +3).

 

4)  Contrapositive of:  "if n is even, then 4 doesn't divide n2 - 1"  is " if 4 doesn't divide n2 - 1, then n is even".

 

5)  4 divides n2 - 1 if and only if n is odd.

 Apr 22, 2020

32 Online Users

avatar
avatar
avatar
avatar