Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

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

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 “int” function on a calculator to calculate $q\text{.}$ [The “int” function is the “greatest integer function.” If $x$ is a real number, then $\mathrm{int}(x)$ is the greatest integer that is less than or equal to $x\text{.}$]

So, in the context of the Division Algorithm, $q=\mathrm{int}\left({\displaystyle \frac{a}{b}}\right)\text{.}$ Consequently,

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

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.