7.E: Exercises
- Page ID
- 7611
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
Convert \(101101101_2\) to base \(10\).
- Answer
-
TBD.
Exercise \(\PageIndex{2}\)
The tripletown number system use only numbers which are \(1\) more than some multiples of \(3\).
- Construct a \(5\times5\) tripletown number system of multiplication table with the numbers \(1,4,7,10\) and \(13.\)
- Find the smallest ten prime numbers in the tripletown number system.
- Find a number with two different tripletown number system prime factorizations.
- Does the Prime divisibility Theorem hold for the tripletown number system? Explain.
Exercise \(\PageIndex{3}\)
The Quadritown number system use only numbers which are \(1\) more than some multiples of \(4\).
- Construct a \(5\times5\) Quadritown number system of multiplication table with the numbers \(1,5,9,13\) and \(17.\)
- Find the smallest ten prime numbers in the Quadritown number system.
- Find a number with two different Quadritown number system prime factorizations.
- Does the Prime divisibility Theorem hold for the Quadritown number system? Explain.
- Answer
-
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Construct a 5X5 Quadritown multiplication table with the numbers 1, 5, 9, 13 and 17.
X
1
5
9
13
17
1
1
5
9
13
17
5
5
25
45
65
85
9
9
45
81
117
153
13
13
65
117
169
221
17
17
85
153
221
289
2. Find the smallest ten prime numbers in Quadritown.
Answer
5, 9, 13, 17, 21, 29, 33, 37, 41, 49
3. Find a number with two different Quadritown prime factorizations.
A solution: (found by trial and error)
Examples
Product
Prime Factorization #1
Prime Factorization
#2
441
(21)(21)
(9)(49)
1089
(33)(33)
(9)(121)
2205
(5)(21)(21)
(5)(9)(49)
3249
(57)(57)
(9)(361)
Does the Prime divisibilty Theorem hold for the Quadritown number system? Explain.
Prime divisibility is defined as follows:
Let p be a prime and let a and b be integers. If p ∣(ab)then p∣a or p∣b.
Thus, prime divisibility does not hold for the Quadritown number system since 21 | 441and 21 | (9)(49), but 21 ∤ 9nor does 21 ∤ 49. Note, this argument could be modified for each of the examples identified in part c.
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