Odd Integer and Multiple of Four

[nelson98]

Homework Statement

Suppose that n is an odd integer. Prove that n is either one greater than a multiple of 4 or one less than a multiple of 4.

Homework Equations

N/A

The Attempt at a Solution

I realize that this is going to be a direct proof. However, I am stumped on where to go from here.

 

[JonF]

We know that the if n is a odd integer it is in form of 2a+1.

Do two cases:
Case 1 a is even so a = 2b for some b
Case 2: a is odd so a = 2b+1 for some b (hint: express 3 in terms of 4)

 

[HallsofIvy]

A slightly different way: any integer must be of one of these forms:
a) 4n
b) 4n+ 1
c) 4n+ 2
d) 4n+ 3
for some n. Both 4n= 2(2n) and 4n+2= 2(2n+1) are even. Can you show that 4n+ 3= 4m- 1 for some number m?