Vengeance Sound MEGA PACK 092012torrent
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Vengeance Sound MEGA PACK (09.2012).torrent
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Vengeance Sound MEGA PACK (09.2012).torrent – DOWNLOAD (Mirror #1)
Latest or the newest version? Specs & Description for Windows 7 x64 (or Vista).. ® Vengeance Sound MEGA PACK .Q:
Prove that for any prime $p$ dividing $n$, $p\mid n$ (or $p$ is a factor of $n$)
I have to prove that for any prime $p$ dividing $n$, $p\mid n$ (or $p$ is a factor of $n$).
I’ve already proved the other direction, $p\mid n$ (or $p$ is a factor of $n$) $\Rightarrow p\mid n \Rightarrow p
mid n$, which is very easy to do.
Let $p$ be a prime and $n \in \mathbb{N}$; let $p \mid n$. I want to prove that $p \mid n$.
If $n=p^k$ then I get $p \mid n$ by definition, and the proof is complete.
If $n=p^n$ then I have to prove that $p \mid n$, and I’m not quite sure how to do this. I can’t use the information that $p \mid n$ (or $p$ is a factor of $n$). Can someone help me with the second option?
Thank you in advance.
A:
$$
n=p_1^{a_1}\cdots p_r^{a_r}
$$
Now, if $p$ is a prime divisor of $n$, then
$$
p_1^{a_1}\cdots p_r^{a_r} \ge n
$$
and
$$
p_1^{a_1}\cdots p_r^{a_r}\mid n\iff p_1^{a_1}\cdots p_r^{a_r}=1
$$
Since $p_1\cdots p_r=p$ and $a_1\cdots a_r\ge 1$, we are done.
A:
$\textbf{No. of primes }\pi\geqslant\frac{\pi\cdot\pi}{2}=