--- ## Definition > [!tldr] Definition > If $a$ and $b$ are two [[integers]], then we say **$b$ divides $a$** and write $b | a$ if there exists an [[Integers|integer]] $k$ such that $a = bk$. Notes: - If $b$ divides $a$, we can also say that $a$ is a multiple of $b$. - Note that the notation $a|b$ is not the fraction $a/b$ or $b/a$. The vertical bar is not a fraction bar, but shorthand for a [[Propositions|statement]] about the relationship between $a$ and $b$. - Divisibility is a special case of the [[Division algorithm|Division Algorithm]] in which there is no remainder. That is, if $a$ and $b$ are [[integers]], the [[Division Algorithm]] states that there must be [[integers]] $q$ and $r$ with $0 \leq r < b$ such that $a = bq + r$. If $r = 0$ in this case, then $b | a$. - If an [[Integers|integer]] $b$ *does not* divide the [[Integers|integer]] $a$, then we indicate this by putting a slash through the vertical bar: $b \nmid a$. ## Examples and Non-Examples * $10 | 200$ because $200 = 20(10)$. * However $10 \nmid 25$ because when dividing $25$ by $10$ we get a nonzero remainder. * We can apply the definition of divisibility to situations where long division doesn't readily apply, for example with negative numbers: We see that $-12 | -48$ because $-48 = 4(-12)$. * The number $0$ is divisible by every other [[Integers|integer]]. Let $a$ be any [[Integers|integer]] whatsoever. Then $0 = 0\cdot a$, which by definition says that $a | 0$. (This is true even if $a = 0$ itself!) * However, $0$ does not divide any [[Integers|integer]] except $0$ itself. If $b$ were an [[Integers|integer]] and $0 | b$, then there would have to be another [[Integers|integer]] $q$ such that $b = q \cdot 0$. But if $b$ is nonzero, this is impossible because the right side of that equation always equals $0$. (This is an example of an [[Indirect proof]].) ## Resources