Question on modulus

Question

Is $x|y$ the same as $x \equiv 0\! \mod\!{y}$ ? If not then how should it be written?

Answer 1

That is not accurate. $2 | 4$, but $2 \not\equiv 0 \pmod{4}$. Instead it should be $y \equiv 0 \pmod{x}$.

Answer 2

If $x, y \in Z$ such that $x | y$, then we are saying there is some integer $k$ such that $xk = y$. This means there is some $k$ such that $xk \equiv 0$ (mod $y$). That doesn't necessarily mean $x \equiv 0$ (mod $y$). It just means that mod $y$, $x$ and $k$ are zero divisors (assuming both are non-zero mod $y$).

As an example, take the integers mod $12$. Then $2 | 12$ since $2 \cdot 6 = 12 \equiv 0$ (mod $12$), but neither $2$ nor $6$ are equivalent to $0$ mod $12$. They are zero divisors mod $12$.

Answer 3

They are not the same.

$x|y$ means $\exists k\in\mathbb{N}$ such that $kx=y$.

Wheras $x=0\mod y$ means $\exists q\in\mathbb{N}$ such that $x=qy$.

As an example $5|10$ but $5 \neq 0 \mod 10$ on the other hand $10\not|\;\;5$ but $10=0\mod 5$.

What you might be thinking of is that $x|y$ is the same as $y=0\mod x$.

Answer 4

$x|y$ means that exists a integer number $k_1$ with $y=k_1x$
$x\equiv 0 \mod y$ means that $x=0+k_2y$ and $k_2$ is a integer number