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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.

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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 $r=log_3 n$.

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

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