1.21: Addition and Multiplication in Zm
- Page ID
- 82303
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter we show how to define addition and multiplication of residue classes modulo \(m\). With respect to these binary operations \(\mathbb{Z}_m\) is a ring as defined in Appendix A.
For \([a],[b]\in\mathbb{Z}_m\) we define \[[a]+[b]=[a+b]\nonumber\] and \[[a][b]=[ab].\nonumber\]
Example \(\PageIndex{1}\)
For \(m=5\) we have \[[2]+[3]=[5],\nonumber \] and \[[2][3]=[6].\nonumber \] Note that since \(5\equiv 0\pmod 5\) and \(6\equiv 1\pmod 5\) we have \([5]=[0]\) and \([6]=[1]\) so we can also write \[\begin{aligned} \left[2\right]+[3]&=[0] \\ [2][3]&=[1].\end{aligned}\]
Since a residue class can have many representatives, it is important to check that the rules given in Definition \(\PageIndex{1}\) do not depend on the representatives chosen. For example, when \(m=5\) we know that \[[7]=[2]\text{ and }[11]=[21]\nonumber \] so we should have \[[7]+[11]=[2]+[21]\nonumber \] and \[[7][11]=[2][21].\nonumber \] In this case we can check that \[[7]+[11]=[18]\text{ and }[2]+[21]=[23].\nonumber \] Now \(23\equiv 18\pmod 5\) since \(5\mid 23-18\). Hence \([18]=[23]\), as desired. Also \([7][11]=[77]\) and \([2][21]=[42]\). Then \(77-42=35\) and \(5\mid 35\) so \(77\equiv 42\pmod 5\) and hence \([77]=[42]\), as desired.
For any modulus \(m>0\) if \([a]=[b]\) and \([c]=[d]\) then \[[a]+[c]=[b]+[d]\nonumber \] and \[[a][c]=[b][d].\nonumber \]
- Proof
-
(This follows immediately from Theorem 15.3 and Theorem 19.2.)
Exercise \(\PageIndex{1}\)
Prove Theorem \(\PageIndex{1}\).
When performing addition and multiplication in \(\mathbb{Z}_m\) using the rules in Definition \(\PageIndex{1}\), due to Theorem \(\PageIndex{1}\), we may at any time replace \([a]\) by \([a']\) if \(a\equiv a'\pmod m\). This will sometimes make calculations easier.
Example \(\PageIndex{2}\)
Take \(m=151\). Then \(150\equiv-1\pmod{151}\) and \(149\equiv-2\pmod{151}\), so \[[150][149]=[-1][-2]=[2]\nonumber \] and \[[150]+[149]=[-1]+[-2]=[-3]=[148]\nonumber \] since \(148\equiv-3\pmod{151}\).
When working with \(\mathbb{Z}_m\) it is often useful to write all residue classes in the least nonnegative residue system, as we do in constructing the following addition and multiplication tables for \(\mathbb{Z}_4\).
Table \(\PageIndex{1}\)
| \(+\) | \([0]\) | \([1]\) | \([2]\) | \([3]\) |
|---|---|---|---|---|
| \([0]\) | \([0]\) | \([1]\) | \([2]\) | \([3]\) |
| \([1]\) | \([1]\) | \([2]\) | \([3]\) | \([0]\) |
| \([2]\) | \([2]\) | \([3]\) | \([0]\) | \([1]\) |
| \([3]\) | \([3]\) | \([0]\) | \([1]\) | \([2]\) |
Table \(\PageIndex{2}\)
| \(\cdot\) | \([0]\) | \([1]\) | \([2]\) | \([3]\) |
|---|---|---|---|---|
| \([0]\) | \([0]\) | \([0]\) | \([0]\) | \([0]\) |
| \([1]\) | \([0]\) | \([1]\) | \([2]\) | \([3]\) |
| \([2]\) | \([0]\) | \([2]\) | \([0]\) | \([2]\) |
| \([3]\) | \([0]\) | \([3]\) | \([2]\) | \([1]\) |
Recall that by Exercise 15.1 we have for all \(a\) and \(m>0\) \[a\equiv a\bmod m\pmod m.\nonumber \] So using residue classes modulo \(m\) this gives \[[a]=[a\bmod m].\nonumber \]
Hence,
\[ [a]+[b]=[(a+b)\bmod m]\nonumber\]
\[[a][b]=[(ab)\bmod m]\nonumber \]
So if \(a\) and \(b\) are in the set \(\{0,1,\dotsc,m-1\}\), these equations give us a way to obtain representations of the sum and product of \([a]\) and \([b]\) in the same set. This leads to an alternative way to define \(\mathbb{Z}_m\) and addition and multiplication in \(\mathbb{Z}_m\). For clarity we will use different notation.
Definition \(\PageIndex{2}\)
For \(m>0\) define \[J_m=\{0,1,2,\dotsc,m-1\}\nonumber \] and for \(a,b\in J_m\) define \[\begin{aligned} a\oplus b &=(a+b)\bmod m \\ a\odot b &=(ab)\bmod m.\end{aligned}\]
Remark \(\PageIndex{1}\)
\(J_m\) with \(\oplus\) and \(\odot\) as defined is isomorphic to \(\mathbb{Z}_m\) with addition and multiplication given by Definition \(\PageIndex{1}\). [Students taking Elementary Abstract Algebra will learn a rigorous definition of the term isomorphic. For now, we take “isomorphic” to mean “has the same form.”] The addition and multiplication tables for \(J_4\) are:
Table \(\PageIndex{3}\)
| \(\oplus\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|
| \(0\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \(1\) | \(1\) | \(2\) | \(3\) | \(0\) |
| \(2\) | \(2\) | \(3\) | \(0\) | \(1\) |
| \(3\) | \(3\) | \(0\) | \(1\) | \(2\) |
Table \(\PageIndex{4}\)
| \(\odot\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|
| \(0\) | \(0\) | \(0\) | \(0\) | \(0\) |
| \(1\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \(2\) | \(0\) | \(2\) | \(0\) | \(2\) |
| \(3\) | \(0\) | \(3\) | \(2\) | \(1\) |
Exercise \(\PageIndex{2}\)
Prove that for every modulus \(m>0\) we have for all \(a,b\in J_m\) \[[a]+[b]=[a\oplus b],\nonumber \] and \[[a][b]=[a\odot b].\nonumber \]
Exercise \(\PageIndex{4}\)
Without doing it, tell how to obtain addition and multiplication tables for \(\mathbb{Z}_5\) from the work in Exercise \(\PageIndex{3}\).
Example \(\PageIndex{3}\)
Let’s solve the congruence \[\label{eq:1} 272x\equiv 901\pmod 9. \] Using residue classes modulo \(9\) we see that (1) is equivalent to \[\label{eq:2} [272x]=[901] \] which is equivalent to \[\label{eq:3} [272][x]=[901] \] which is equivalent to \[\label{eq:4} [2][x]=[1]. \] Now we know \([x]\in\{[0],[1],\dotsc,[8]\}\) so by trial and error we see that \(x=5\) is a solution.