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Beginning Activity Beginning Activity 2: The Remainder When Dividing by 9
A4US

If \(a\) and \(b\) are integers with \(b > 0\text{,}\) then from the Division Algorithm, we know that there exist unique integers \(q\) and \(r\) such that

\begin{equation*} a = bq + r \text{ and } 0 \leq r \lt b\text{.} \end{equation*}

In this activity, we are interested in the remainder \(r\text{.}\) Notice that \(r = a - bq\text{.}\) So, given \(a\) and \(b\text{,}\) if we can calculate \(q\text{,}\) then we can calculate \(r\text{.}\)

We can use the ā€œintā€ function on a calculator to calculate \(q\text{.}\) [The ā€œintā€ function is the ā€œgreatest integer function.ā€ If \(x\) is a real number, then \(\operatorname{int}( x )\) is the greatest integer that is less than or equal to \(x\text{.}\)]

So, in the context of the Division Algorithm, \(q = \operatorname{int} \!\left( {\dfrac{a}{b}} \right)\text{.}\) Consequently,

\begin{equation*} r = a - b \cdot \operatorname{int} \!\left( {\frac{a}{b}} \right)\text{.} \end{equation*}

If \(n\) is a positive integer, we will let \(s\left( n \right)\) denote the sum of the digits of \(n\text{.}\) For example, if \(n = 731\text{,}\) then

\begin{equation*} s( {731} ) = 7 + 3 + 1 = 11\text{.} \end{equation*}

For each of the following values of \(n\text{,}\) calculate

  • The remainder when \(n\) is divided by 9, and

  • The value of \(s( n )\) and the remainder when \(s( n )\) is divided by 9.

1.

\(n=498\)

2.

\(n=7319\)

3.

\(n=4672\)

4.

\(n=9845\)

5.

\(n=51381\)

6.

\(n=305877\)

What do you observe?