4.6: Section 6-Exercises
- Page ID
- 33712
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Use the method of direct proof to prove the following statements.
Exercise \(\PageIndex{1}\)
If x is an even integer, then \(x^2\) is even.
Exercise \(\PageIndex{2}\)
If x is an odd integer, then \(x^3\) is odd.
Exercise \(\PageIndex{3}\)
If a is an odd integer, then \(a^2+3a+5\) is odd.
Exercise \(\PageIndex{4}\)
Suppose \(x,y \in \mathbb{Z}\). If x and y are odd, then xy is odd.
Exercise \(\PageIndex{5}\)
Suppose \(x, y \in \mathbb{Z}\). If x is even, then xy is even.
Exercise \(\PageIndex{6}\)
Suppose \(a, b, c \in \mathbb{Z}\). If \(a|b\) and \(a|c\), then \(a|(b+c)\).
Exercise \(\PageIndex{7}\)
Suppose \(a, b \in \mathbb{Z}\). If \(a|b\), then \(a^{2}|b^{2}\).
Exercise \(\PageIndex{8}\)
Suppose a is an integer. If \(5|2a\), then \(5|a\).
Exercise \(\PageIndex{9}\)
Suppose a is an integer. If \(7|4a\), then \(7|a\).
Exercise \(\PageIndex{10}\)
Suppose a and b are integers. If \(a|b\), then \(a|(3b^3-b2+5b)\).
Exercise \(\PageIndex{11}\)
Suppose \(a, b, c, d \in \mathbb{Z}\). If \(a|b\) and \(c|d\), then \(ac|bd\).
Exercise \(\PageIndex{12}\)
If \(x \in \mathbb{R}\) and \(0<x<4\), then \(\frac{4}{x(4−x)} \ge 1\).
Exercise \(\PageIndex{13}\)
Suppose \(x, y \in \mathbb{R}\). If \(x^2+5y = y^2+5x\), then \(x=y\) or \(x+y = 5\).
Exercise \(\PageIndex{14}\)
If \(n \in \mathbb{Z}\), then \(5n^2+3n+7\) is odd. (Trycases.)
Exercise \(\PageIndex{15}\)
If \(n \in \mathbb{Z}\), then \(n^2+3n+4\) is even. (Trycases.)
Exercise \(\PageIndex{16}\)
If two integers have the same parity, then their sum is even. (Try cases.)
Exercise \(\PageIndex{17}\)
If two integers have opposite parity, then their product is even.
Exercise \(\PageIndex{18}\)
Suppose x and y are positive real numbers. If \(x < y\), then \(x^{2} < y^{2}\).
Exercise \(\PageIndex{19}\)
Suppose a, b and c are integers. If \(a^{2}|b\) and \(b^{3}|c\), then \(a^6|c\).
Exercise \(\PageIndex{20}\)
If a is an integer and \(a^{2}|a\), then \(a \in \{−1, 0, 1\}\).
Exercise \(\PageIndex{21}\)
If p is prime and k is an integer for which \(0<k<p\), then p divides \({p \choose k}\).
Exercise \(\PageIndex{22}\)
If \(n \in \mathbb{N}\), then \(n^{2} = 2{n \choose 2} + {n \choose 1}\). (You may need a separate case for \(n = 1\).)
Exercise \(\PageIndex{23}\)
If \(n \in \mathbb{N}\), then \({2n \choose n}\) is even.
Exercise \(\PageIndex{24}\)
If \(n \in \mathbb{N}\) and \(n \ge 2\), then the numbers \(n!+2, n!+3, n!+4, n!+5, \cdots , n!+n\) are all composite. (Thus for any \(n \ge 2\), one can find \(n - 1\) consecutive composite numbers. This means there are arbitrarily large “gaps” between prime numbers.)
Exercise \(\PageIndex{25}\)
If \(a, b, c \in \mathbb{N}\) and \(c \le b \le a\), then \({a \choose b}{a \choose c} = {a \choose b-c}{a-b+c \choose c}\).
Exercise \(\PageIndex{26}\)
Every odd integer is a difference of two squares. (Example \(7 = 4^{2}-3^{2}\), etc.)
Exercise \(\PageIndex{27}\)
Suppose \(a, b \in \mathbb{N}\). If \(gcd(a, b) > 1\), then \(b|a\) or b is not prime.
Exercise \(\PageIndex{28}\)
Let \(a, b, c \in \mathbb{Z}\). Suppose a and b are not both zero, and \(c \ne 0\). Prove that \(c \cdot gcd(a,b) \le gcd(ca, cb)\).