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Question

"If we find a prime $p$ such that $p\mid n$ , then $n/p$ is a positive integer that's smaller than $n$."

I understand $n/p$ is $n$ divided by $p$ but what is $n\mid p$?

Answer 1

For nonzero integers $x$ and $y$, the following are equivalent statements:

• $x$ divides $y$

• $x\mid y~~~~~~~~~~~~$(this is read aloud the same way as the previous line)

• $y$ is divisible by $x$

• $y\equiv 0\pmod{x}$

• $\frac{y}{x}$ is an integer

• There exists some integer $k$ for which $xk = y$

• $y$ is a multiple of $x$

• $x$ is a factor of $y$

On the other hand, $x/y$ is in this context another way of writing the fraction $\frac{x}{y}$

Answer 2

$p \mid n$ means "p divides n" or $\exists k\in\mathbb{Z}$ so $n=pk$

Answer 3

The expression "$n/p$" represents the number that you get when you divide $n$ by $p$, whereas the expression "$p\mid n$" represents the statement that $p$ divides evenly into $n$.