Is $30$ a factor of $n$?

Question

Let $n$ be an integer and assume that $30$ is a factor of $n^2$ and $15$ is a factor of $n$. Prove that $30$ is a factor of $n$.

I tried testing some numbers, e.g., $n = 30$ clearly works since $15|30$ and $30|30^2$. Also $n = 60$ works. But, how can I prove this generally?

Answer 1

1. Show that $2$ is a factor of $n^2$.
2. From that, show that $2$ is also a factor of $n$.
3. You then know that $15$ is a factor of $n$, and $2$ is a factor of $n$, and you should then easily be able to show that $30$ is a factor of $n$.

Answer 2

This actually works more broadly. Prime numbers are defined in the following way:

We say that $p$ is prime if whenever $p$ divides $a \cdot b$ (written $p \mid a \cdot b$), then $p$ divides $a$ or $p$ divides $b$.

In particular, if $p \mid n^2$, then it follows that $p \mid n$.

Now, note that $30 = 2 \cdot 3 \cdot 5$ is a product of primes (none of which are to a higher power than one). It follows that if $30 \mid n^2$, then $2 \mid n^2$, $3 \mid n^2$, and $5 \mid n^2$. Hence...