1.17: Divisibility Tests for 7 and 13
- Page ID
- 82299
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let \(a=a_ra_{r-1}\dotsm a_1a_0\) be the decimal representation of \(a\). Then
- \(7\mid a\Leftrightarrow 7\mid a_r\dotsm a_1-2a_0\).
- \(13\mid a\Leftrightarrow 13\mid a_r\dotsm a_1-9a_0\).
[Here \(a_r\dotsm a_1=\frac{a-a_0}{10}=a_r10^{r-1}+\dotsb+a_210+a_1\).]
- Proof
-
Proof of (a). Let \(c=a_r\dotsm a_1\). So we have \(a=10c+a_0\). Hence \(-2a=-20c-2a_0\). Now \(1\equiv -20\pmod 7\) so we have \[-2a\equiv c-2a_0\pmod 7.\nonumber \] It follows from Theorem 15.1 that \[-2a\bmod 7=c-2a_0\bmod 7.\nonumber \] Hence, \(7\mid -2a\Leftrightarrow 7\mid c-2a_0\). Since \(\gcd(7,-2)=1\) we have \(7\mid -2a\Leftrightarrow 7\mid a\). Hence \(7\mid a\Leftrightarrow 7\mid c-2a_0\), which is what we wanted to prove.
Proof of (b). (This has a similar proof to that for \(\PageIndex{1}\) (a) and is left for the interested reader.)
Before proving this theorem we illustrate it with two examples. \[\begin{split} 7\mid 2481 &\Leftrightarrow 7\mid 248-2 \\ &\Leftrightarrow 7\mid 246 \\ &\Leftrightarrow 7\mid 24-12 \\ &\Leftrightarrow 7\mid 12 \end{split}\] since \(7\nmid 12\) we have \(7\nmid 2481\).
\[\begin{split} 13\mid 12987 &\Leftrightarrow 13\mid 1298-63 \\ &\Leftrightarrow 13\mid 1235 \\ &\Leftrightarrow 13\mid 123-45 \\ &\Leftrightarrow 13\mid 78 \end{split}\] since \(6\cdot 13=78\), we have \(13\mid 78\). So, by Theorem \(\PageIndex{1}\) (b), \(13\mid 12987\).
Exercise \(\PageIndex{1}\)
Use Theorem \(\PageIndex{1}\) (a) to determine which of the following are divisible by \(7\):
- \(6994\)
- \(6993\)
Exercise \(\PageIndex{2}\)
In the notation of Theorem \(\PageIndex{1}\), show that \(a\bmod 7\) need not be equal to \((a_r\dotsm a_1-2a_0) \bmod 7.\).