# Homework Help: Consecutive odd natural numbers - one is composite. Prove!

1. Sep 29, 2010

### nelson98

1. The problem statement, all variables and given/known data

Every triple of consecutive odd natural numbers, with the first being at least 5, contains at least on composite.

2. Relevant equations

N/A

3. The attempt at a solution

I know from number theory that of every set of consecutive odd integers, one of them is divisible by three, thereby making it a composite number. I just don't know how to prove it. Can anybody help?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 29, 2010

### HallsofIvy

A triple of consectutive odd integers can be written as 2n+1, 2n+ 3, and 2n+ 5.

Suppose 2n+ 1 is NOT a multiple of 3. Then it has a remainder of either 1 or 2 when divided by 3

a) Suppose 2n+ 1 has remainder 1 when divided by 3: 2n+ 1= 3k+ 1 so that 2n= 3k. Look at both 2n+3 and 2n+ 5 in this case.

b) Suppose 2n+ 1 has remainder 2 when divided by 3: 2n+ 1= 3k+ 2 so that 2n= 3k+ 1. Look at both 2n+3 and 2n+ 5 in this case.