8.4: Homework
- Page ID
- 70331
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Submit homework separately from this workbook and staple all pages together. (One staple for the entire submission of all the unit homework)
- Start a new module on the front side of a new page and write the module number on the top center of the page.
- Answers without supporting work will receive no credit.
- Some solutions are given in the solutions manual.
- You may work with classmates but do your own work.
Find GCF(252, 350) using:
| a. prime factorization; |
| b. Old Chinese Method |
| c. Euclidean Algorithm |
| d. Compute LCM(252,350) |
Find GCF(140, 315) using:
| a. prime factorization; |
| b. Old Chinese Method |
| c. Euclidean Algorithm |
| d. Compute LCM(252,350) |
Use the Euclidean Algorithm to compute the greatest common factor of the numbers given. Use correct notation, and show each step. Then, show how you check your answer. Also, compute the LCM of the two numbers.
|
a. GCF(3525, 658) LCM(3525, 658) |
|
b. GCF(1075, 1548) LCM(1075, 1548) |
| a. If 6|482__354 What digits could go on the blank? Explain. |
| b. If 6|482354__ What digits could go on the blank? Explain. |
State whether each of the following statements is true or false. If it is false, provide a counterexample. If it is true, provide an example.
a. If (a + b)|c, then a|c and b|c
b. If a|b and a|c, then a|(bc)
c. If a|b and a|(b + c), then a|c
d. If a|bc, then a|b and a|c
e. If a|b and a|c, then a|(b + c)
Write the prime factorization for the following numbers. If it is prime, write "prime" and explain how you know it is prime.
| a. 371 | b. 429 | c. 197 | d. 287 |
Assume m and n are composite whole numbers in each of the following. Find the following. Then provide an example using numbers for m (and n where used). Remember not to use prime numbers in your example.
a. GCF(m,m) =
b. LCM(m,m) =
c. GCF(m,0) =
d. GCF(m,1) =
e. If GCF(m,n) = 1, then LCM(m,n) =
f. If GCF(m,n) = m, then LCM(m,n) =
g. If LCM(m,n) = mn, then GCF(m,n) =
| a. Formally prove that the sum of two odd numbers is even. |
| b. Formally prove that the product of two odd numbers is odd. |
Find the following sums using methods from this module: Show all work
a. 1 + 2 + 3 + . . . + 313 + 314 + 315 =
b. 111 + 112 + 113 + . . . + 287 + 288 + 289 =
c. 15 + 30 + 45 + . . . + 900 + 915 + 930 =
d. 102 + 105 + 108 + . . . + 300 + 303 + 306 =
On each number line, state all whole number possibilities less than 100 that the man could be standing on.
|
a.  |
|
b.  |
|
c.  |
|
d.  |
|
e.  |
The factors of a number that are less than the number itself are called proper factors. For instance, the proper factors of 10 are 1, 2 and 5. A number is classified as deficient if the sum of its proper factors is less than the number itself. 10 is a deficient number since 1 + 2 + 5 < 10. A number is classified as abundant if the sum of its proper factors is greater than the number itself. For instance, the proper factors of 18 are 1, 2, 3, 6, and 9. 18 is a deficient number since 1 + 2 + 3 + 6 + 9 > 18. A number is classified as perfect if the sum of its proper factors equals the number itself. For each number, list its proper factors. Then find the sum of its proper factors. Then, classify each number as deficient, abundant or perfect.
| a. Proper factors of 6: ___________ ; Sum: _____ ; Classification: |
| b. Proper factors of 7: ___________ ; Sum: _____ ; Classification: |
| c. Proper factors of 8: ___________ ; Sum: _____ ; Classification: |
| d. Proper factors of 9: ___________ ; Sum: _____ ; Classification: |
| e. Proper factors of 11: ___________ ; Sum: _____ ; Classification: |
| f. Proper factors of 12: ___________ ; Sum: _____ ; Classification: |
Are prime numbers deficient, perfect, or abundant? ________ Explain why.
Answer true or false for each of the following. If it is true, provide an example. If it is false, provide a counterexample.
a. Every prime number is odd.
b. If a number is divisible by 6, then it is divisible by 2 and 3.
c. If a number is divisible by 2 and 6, then it is divisible by 12.
d. If a number is divisible by 3 and 4, then it is divisible by 12.
e. If a \(\neq\) b, then GCF(a, b) < LCM(a, b).
f. If 6 is a factor of mn, then 6 is a factor of m or a factor of n.
g. If 5 is a factor of mn, then 5 is a factor of m or a factor of n.
Can the sum of two odd prime numbers be a prime number? Explain why or why not.
Find the least common multiple of the following sets of numbers:
a. LCM(2, 4, 5, 7, 8, 12, 14, 15)
b. LCM(3, 4, 6, 8, 9, 10, 12, 18)
If GCF(30, x) = 6 and LCM(30, x) = 180, then what is x? (Hint: see page 65)
The theory of biorhythm states that your physical cycle is 23 days long, your emotional cycle is 28 days long, your intellectual cycle is 33 days long. If your cycles all occur on the same day, how many days until your cycles again occur on the same day? About how many years is this?