Division with remainder

Numbers: Integers

Division with remainder

Thus far the operation "division" has not yet been addressed. One reason for this is that division of an integer (that is unequal to $0$ ) on another integer does not always produce an integer.

An integer $b$ is called a divisor of an integer $a$ if $a$ is a whole multiple of $b$ ; in other words, there is an integer $q$ such that $a=b\times q$ .

In this case the number is $q$ uniquely determined. It is called the quotient of $a$ when divided by $b$ , and its notation is $\frac{a}{b}$ or (less frequent) $a:b$ .

Instead of " $b$ is a divisor of $a$ " we also say " $b$ divides $a$ " or " $a$ can be divided by $b$ ".

The priority of division is equal to the priority of multiplication. In an expression containing only multiplications and divisions, the precedence is taken from left to right.

The notation $\frac{a}{b}$ corresponds to the usual notation of a rational number. The meaning of $\frac{a}{b}$ if $b$ cannot divided by $a$ will be discussed later.

It will become clear that laws as $\frac{-a}{b} = -\frac{a}{b} = \frac{a}{-b}$ and $\frac{a+c}{b}=\frac{a}{b}+\frac{c}{b}$ are true if $a$ and $c$ both are integer multiples of $b$ .

The priority rules for division are clarified by the following formulas, where $a$ , $b$ , $c$ , and $d$ are integers such that $b$ divides $a$ and $d$ divides $c$ :

$\begin{array}{rcl}a: b\cdot c &=& \left(a: b\right)\cdot c\\ a: b\cdot c: d &=& \left(\left(a: b\right)\cdot c\right): d\\ a: b + c: d &=& \left(a: b\right)+ \left(c: d\right)\\ a: b - c: d &=& \left(a: b\right)- \left(c: d\right)\\ \end{array}$

If $\frac{a}{b}$ is not an integer, for example when $a=4$ and $b=3$ , then you can still determine how many multiples of $b$ fit in $a$ . This number of multiples, and what remains after subtraction of these multiples of $a$ , are called quotient and remainder, respectively. For example, $5$ is the quotient and $3$ is the remainder of division with remainder of $4113$ by $822$ :

This gives the equality ${4113}=5\times822+{3}$ , with $0\le 3\lt 822$ .

Division with remainder

If $a$ is a non-negative integer and $b$ natural number, then there is exactly one integer $q$ and one integer $r$ such that

$a=q\times b+r \phantom{xxx} \text{ and }\phantom{xxx}0\le r\lt b\tiny.$ Here, $q$ is called the quotient and $r$ the remainder of the division with remainder of $a$ by $b$ .

We write $\left\lfloor \frac{a}{b}\right\rfloor$ for the quotient. The remainder is equal to $a-\left\lfloor \frac{a}{b}\right\rfloor\times b$ .

The number $a$ is divisible by $b$ if and only if the remainder of the division of $a$ by $b$ is equal to $0$ .

The notation $\left\lfloor \frac{a}{b}\right\rfloor$ corresponds to the usual notation $\lfloor x\rfloor$ for the greatest integer less than or equal to $x$ , which we will discuss later.

Is the number $170$ a multiple of $-17$ ?

yes

For, $170= -10\times (-17) \text{} {}$ .

New example

Numbers: Integers

Division with remainder

What can the AI Bot help you with?