Numbers: Integers
Divisors of integers
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}=822\times 5+{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.
For, #-39= -3\times 13 \text{} {}#.
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