# Composite numbers with at most one prime factor less than root of it

Let $$n$$ be a composite number so that it has at most one prime factor such that $$p \leq \sqrt{n}$$. Find all ways how you can factor $$n$$ and give examples.

My work:

We know that if $$n$$ is composite, there has to be at least one prime factor with such property. We also can deduce that for any other prime factors of $$n$$ - lets call them $$q_i$$ there can be at most one of them because $$q_i^2 > n$$. So the different ways you could factor $$n$$ I came up with are

1) $$n=p^k$$ - for example $$16=2^4$$. 2) $$n=p^k \cdot q$$ - for example $$44 = 2^2 \cdot 11$$. 3) $$n=p\cdot q$$ - for example $$22 = 2 \cdot 11$$.

Are there any ways that I am missing?

• Looks good to me. Note that your case 3 ($n=p\cdot q$) is just case 2 with $k=1$. – TonyK Mar 3 at 14:46

You already found out that at most one prime factor can exceed $$\sqrt{n}$$ (this is true for every positive integer $$n$$). Moreover, we know that we must have exactly one prime factor $$p\le \sqrt{n}$$ (Because of the mentioned condition).
If no prime factor exceeeds $$\sqrt{n}$$, then we must have a prime power.
If one prime factor exceeds $$\sqrt{n}$$, the exponent corresponding to this prime must be $$1$$, so we must have a prime power multiplied with this prime.