Complete the following proof.

1. Claim: For any integer,+ + is always odd.

Proof// Letbe an integer. [We wish to show that2+ + 3is odd.]

We know that eitheris even or it is odd. So there are two cases to consider.

(case 1) Suppose thatis even. [We need to show that2+ + 3is odd.]

Sinceis even we may write= __________ where Z.

Now observe that2+ + 3=[_____________]2+ [__________ ] + 3

=42+2+2+1=2(__________)+1. Since the integers are closed under multiplication and addition,

we see that

______________ is an integer. This means that2+ + 3is odd by definition of odd in this case.

(case2) Suppose thatis odd. [We need to show that2 ++ 3 is odd.]

Sinceis odd we may write= __________ where Z.

Now observe that2+ + 3=[_____________]2+ [__________ ] + 3

 =___________________ 

= 2 (______________ ) + 1.

Since the integers are closed under multiplication and addition, we see that

______________ is an integer. This means that2+ + 3is odd by definition of odd in this case. In

both cases we showed that for any integer,2+ + 3is always odd.