Is a composite number always either a product of two coprimes, or a power of a prime itself?

Question

Is a composite number n always either

• a product of two coprimes (i.e. $n = n_1 \cdot n_2$ such that $n_1$ and $n_2$ are coprimes)
• or a power of a prime itself (i.e. $n = x ^ y$ where $x$ is a prime)

?

I would think this is true since when you factorize $n$, if $n$ have two or more prime factors then those prime factors can be arranged to make $2$ coprimes. Otherwise if $n$ can only be factorized using one prime then it can't be the product of two coprimes.

Is this true? Is there any error in my reasoning?

Answer 1

Nothing is wrong with your reasoning. To make it clearer, there are two cases:

1. There is only one prime that divides $n$.

2. There are more than one prime that divides $n$. Let $p$ be a prime dividing $n$. And let $k$ be the largest natural number such that $p^k$ divides $n$. Then $n = p^k m$ for some $m \ne 1$ because some prime divides $n$ but not $p^k$. Also $\gcd(p^k,m) = 1$, so we are done.