So we have a number $n$ which is composite and odd at the same time.

And task was to prove that all of its prime factors must be at most $\frac{n}{3}$.

Second task was to justify how many prime factors the number $n$ could have.

My attempt: I'm not sure if it's even necessary but I know by definition of a composite number $n=km$ for some $1 <k,m<n.$ And $n=2l+1$ for some $l\in\mathbb{N}$.

I also know that this theorem (I don't think it has a name) that for this number $n$ which we know that $n>1$ has a prime factor $p$ such that $p|n$.

I have just begun studying these sorts of topic. Please help, thanks in advance.


Since $n$ is odd, $2$ is not a factor of $n$, so the least prime factor of $n$ could be $3$. Therefore, any prime factor of $n$ could not be more than $n/3$.

To maximize the number of prime factors of $n$, minimize the size of each prime factor, i.e., choose $3$ as each prime factor, i.e., make $n=3^r$, so $n$ has $r$ prime factors, where $r=log_3 n$.

  • $\begingroup$ Hi did you have a chance to read my last comment, and thanks lots! $\endgroup$ – javacoder Jan 24 at 5:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.