Prime factors of odd composites

[GlassBones]

Homework Statement

Let $n$ be odd and a composite number, prove that all of its prime is at most $\frac{n}{3}$

Homework Equations

Some theorems might help?
Any $n>1$ must have a prime factor
if n is composite then there is a prime $p<√n$ such that $p|n$

The Attempt at a Solution

I'm at a standstill, not sure how to start off this question? Can anyone prove hints to get me going

 

[fresh_42]

Homework Statement

Let $n$ be odd and a composite number, prove that all of its prime is at most $\frac{n}{3}$

Homework Equations

Some theorems might help?
Any $n>1$ must have a prime factor
if n is composite then there is a prime $p<√n$ such that $p|n$

The Attempt at a Solution

I'm at a standstill, not sure how to start off this question? Can anyone prove hints to get me going

Are you allowed to use the prime decomposition of integers?

 

[GlassBones]

I'm pretty sure I can (prof didn't talk about it but I'm assuming we should know about it).
With prime decomposition, I'm not sure how to generate it. Can we just assume there is a prime decomposition because n is composite?

 

[fresh_42]

I'm pretty sure I can (prof didn't talk about it but I'm assuming we should know about it).
With prime decomposition, I'm not sure how to generate it. Can we just assume there is a prime decomposition because n is composite?

The fundamental theorem of arithmetic says, that every integer can be written as a product of primes. If you are allowed to use this, and proofs can easily be found, e.g. on Wikipedia, then you only have to write this in a formula and think about what the smallest prime in there is.

 

[GlassBones]

Since n is odd composite, the smallest prime must be 5. So n is divisible by 5.
I see where this is going, but ill probably go over the proof again to make sure I get it

 

[fresh_42]

Start with the replacement of $5$ by $3$ in your argument. You don't need the fundamental theorem of arithmetic, but you have somehow to connect $n$ with primes. A prime number $p$ has the property, that if it divides a composite number, then it divides a factor. This can also be used. It is called Euclid's lemma, but it is actually the correct definition of a prime.

 

[GlassBones]

Oh for some reason I thought the only odd composite numbers was divisible by 5. Not sure why I was thinking that. Thanks