# Homework Help: Prove by Contradiction: For all integers x greater than 11

1. Aug 6, 2012

### Animuo

1. The problem statement, all variables and given/known data
Prove by Contradiction: For all integers x greater than 11, x equals the sum of two composite numbers.

2. Relevant equations
A composite number is any number that isn't prime
To prove by contradiction implies that if you use a statement's as a negation, a contradiction arises

3. The attempt at a solution
The negation of the original statement is:
There exists an integer x such that if x > 11, then x does not equal the sum of two composite numbers.

I'm really stuck on this one, I tried substituting in values using the quotient remainder theorem (a number can be represented as 2r, or 2r + 1.. or alternatively 3r, 3r + 1, or 3x + 2, but I wasn't getting anywhere with it). Some direction would be appreciated!

2. Aug 6, 2012

### micromass

Given that x is not the sum of two composite numbers, what can you say about things like

x-4
x-6
x-8
x-9

3. Aug 7, 2012

### Millennial

All even numbers are composite (besides 2!), and since 4 is composite as well, any even number greater than 4 can be written as the sum of two composite numbers. This leaves only odd ones left.

Can you apply a similar logic for odd numbers?