What does "$x$ divides $y$" mean?

Question

I need to negate the following sentence: "If for the integers $x, y, z$ we know that $x$ divides $y$ and $y$ divides $z$, then $x$ divides $z$."

In this scenario, what does it mean for $x$ to "divide" $y$, et cetera?

Answer 1

If $x$ divides $y$, $x$ is a factor of $y$, or in other words, $y$ is divisible by $x$.

Of course, the prerequisite is that $x,y\in{\Bbb{Z}}$.

For instance, $3$ divides $15$ because $15=3\cdot5$, $3$ is a factor of $15$.

Answer 2

In terms of number theory, let $a,b\in\mathbb{Z}$. Then $a$ divides $b$ if $\exists m\in\mathbb{Z}$ such that $am=b$. This is written $a|b$. For example, $7$ is a divisor of $56$ because $7\times 8=56$. This is the algebraic/proper definition for two elements to divide each other. This satisfies the transitive relation \begin{equation*} a|b~\text{and}~b|c\Rightarrow a|c. \end{equation*} Does this help?

Answer 3

x divides y means there exists an integer n such that nx=y. So 7 divides 28, since 4*7=28, but 8 does not divide 28, even though outside number theory we would happily deal with the number 3.5.

Answer 4

Does it really matter what is the meaning of "$x$ divides $y$"?

This is a binary relation, $D(x,y)$. The statement you need to negate is $\forall x\forall y\forall z(D(x,y)\land D(y,z)\rightarrow D(x,z))$, and negating this statement has nothing to do with its actual content.

In any case, for the integers, $x$ divides $y$ if there exists some integer $k$, such that $x\cdot k=y$. So for example $1$ divides every integer, and every integer divides $0$.

Answer 5

Saying that $x$ divides $y$ means that there exists an integer, say $q$ such that $y=q.x$ (e.g. $4$ divides $12$ with $q=3$). Your assertion is then: $y=q_1 . x$ and $z=q_2 . y$ then $z=(q_1.q_2).x$ and $x|z$ (read $x$ divides $z$ with $q=q_1.q_2$). Is that clear?