To find how often a natural number \(b\) goes into a natural number \(a\text{,}\) we subtract \(b\) from \(a\text{.}\) Then we subtract \(b\) from the result, subtract \(b\) from that result, and continue. We repeat this process until \(b\) no longer goes into the previous result. This happens when the previous result is less than \(b\text{.}\) The number of times we have subtracted \(b\) is called the quotient (of the division of \(a\) by \(b\)). The number left from \(a\) after subtracting \(b\) as often as possible is called the remainder (of the division of \(a\) by \(b\)).
Example3.1.1.How often does \(7\) go into \(26\) ?
To find out how often \(7\) goes into \(26\) we repeatedly subtract \(7\text{.}\) We stop this process when \(7\) no longer goes into the result of the previous subtraction. In other words, we stop when we reach a number that is less than seven. We get:
We stop here because \(7\) does not go into \(5\text{.}\) We have subtracted \(7\) three times. This means that \(7\) goes into \(26\) three times. So the quotient is \(3\text{.}\) We have \(5\) left over. This is the remainder. We have found that
When we divide a positive integer \(a\) by an integer \(b\text{,}\)\(q\) is the number of times \(b\) goes into \(a\text{.}\) So \(q\) is the quotient when \(a\) is divided by \(b\text{.}\) Because \(b\) goes into \(a\) exactly \(q\) times and \(b\) cannot go into \(a\) more than \(q\) times, we have
\begin{equation*}
a-q\cdot b\lt b
\end{equation*}
So the remainder is \(r:=a-q\cdot b\) with \(0\le r \lt b\text{.}\) We have
\begin{equation*}
a=q\cdot b+r.
\end{equation*}
If \(a\) is written in the form \(a=q\cdot b+r\) with \(0\le r \lt b\text{,}\) then it is easy to find the quotient and remainder when \(a\) is divided by \(b\text{.}\)
As \(501656 \lt 577590\) this is of the form \(a=q\cdot b+r\) with \(0\le r \lt b\text{.}\) Now we read off the quotient \(q=490687\) and the remainder \(r=501656\text{.}\)
When we divide a positive integer \(a\) by an integer \(b\) that is greater than \(a\) then \(b\) does not go into \(a\text{.}\) So \(b\) goes into \(a\) zero times and what is left over, the remainder, is \(a\text{.}\)
We write the process above as an algorithm called the Division Algorithm, and then extend it to negative integers. We introduce the operations \(\fdiv\) and \(\fmod\) as notation for quotients and remainders. We also describe how to do integer division with a calculator. We study properties of the operation \(\fmod\) and give an example of using it to validate ISBN numbers.
