Proving that if $n$ divides $p^{\alpha}$, where $p$ is prime and $\alpha \geq 0$, then $n = p^{\beta}$, where $0 \leq \beta \leq \alpha$.

Question

I think this could be done using the fundamental theorem of arithmetic, but I'm not sure I want to use something that strong. How do I go about this?

Answer 1

For a prime number $p$, the only divisors of $p^{\alpha }$ are $\{1,p,p^2,...,p^{\alpha}\}$

Answer 2

A key property of prime numbers is:

If $q$ is prime and $q$ divides $ab$, then $q$ divides $a$ or $q$ divides $b$.

Therefore, by induction, if $q$ divides $p^\alpha$, then $q=p$.

Hence, $p$ is the only possible prime divisor of $n$ and so $n$ is a power of $p$.