# A number with greatest amount of factors that are prime and at the same time finding the upper limit of non-composite factors for a number

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 $$1 And $$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 $$p|n$$.

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 $$r=log_3 n$$.