Appendix C Hints and Answers to Selected Even Exercises

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Use the Fundamental Theorem of Arithmetic.
  </p>

</hint>

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  <p>
    Let <m>S = \{s \in {\mathbb N} : a \mid s</m>, <m>b \mid s \}</m>.
    Then <m>S \neq \emptyset</m>, since <m>|ab| \in S</m>.
    By the Principle of Well-Ordering, <m>S</m> contains a least element <m>m</m>.
    To show uniqueness, suppose that <m>a \mid n</m> and <m>b \mid n</m> for some <m>n \in {\mathbb N}</m>.
    By the division algorithm, there exist unique integers <m>q</m> and <m>r</m> such that <m>n = mq + r</m>, where <m>0 \leq r \lt m</m>.
    Since <m>a</m> and <m>b</m> divide both <m>m</m>, and <m>n</m>, it must be the case that  <m>a</m> and <m>b</m> both divide <m>r</m>.
    Thus, <m>r = 0</m> by the minimality of <m>m</m>.
    Therefore, <m>m \mid n</m>.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Since <m>\gcd(a,b) = 1</m>, there exist integers <m>r</m> and <m>s</m> such that <m>ar + bs = 1</m>.
    Thus, <m>acr + bcs = c</m>.
    Since <m>a</m> divides both <m>bc</m> and itself, <m>a</m> must divide <m>c</m>.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Every prime must be of the form 2, 3, <m>6n + 1</m>, or <m>6n + 5</m>.
    Suppose there are only finitely many primes of the form <m>6k + 5</m>.
  </p>

</hint>

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  <p>

    <md>
      \begin{array}{c|cccc} \cdot &amp; 1  &amp; 5  &amp; 7  &amp; 11 \\ \hline 1 &amp; 1 &amp; 5 &amp; 7 &amp; 11 \\ 5 &amp; 5 &amp; 1 &amp; 11 &amp; 7 \\ 7 &amp; 7 &amp; 11 &amp; 1 &amp; 5 \\ 11 &amp; 11 &amp; 7 &amp; 5 &amp; 1 \end{array}
    </md>

  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Pick two matrices.
    Almost any pair will work.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Look at the symmetry group of an equilateral triangle or a square.
  </p>

</hint>

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  <p>
    Let
    <md>
      \sigma = \begin{pmatrix} 1 &amp; 2 &amp; \cdots &amp; n \\ a_1 &amp; a_2 &amp; \cdots &amp; a_n \end{pmatrix}
    </md>

    be in <m>S_n</m>.
    All of the <m>a_i</m>s must be distinct.
    There are <m>n</m> ways to choose <m>a_1</m>, <m>n-1</m> ways to choose <m>a_2</m>, <m>\ldots</m>, 2 ways to choose <m>a_{n - 1}</m>, and only one way to choose <m>a_n</m>.
    Therefore, we can form <m>\sigma</m> in <m>n(n - 1) \cdots 2 \cdot 1 = n!</m> ways.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Look at <m>S_3</m>.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    What did you see last time you went driving?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Maybe go out for a drive?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Do you know the acronym<ellipsis/><acro>ROY G BIV</acro> for the colors of a rainbow, and their order?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    What did you see last time you went driving?
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Maybe go out for a drive?
  </p>

</hint>

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  <p>
    Dorothy will not be much help with this proof.
  </p>

</hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>
    Dorothy will not be much help with this proof.
  </p>

</hint>

                        <hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

                          <p>
Do you really need a hint? Carefully reread the question.
                          </p>

                        </hint>

                            <hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

                              <p>
What did you see last time you went driving?
                              </p>

                            </hint>

                            <hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

                              <p>
Maybe go out for a drive?
                              </p>

                            </hint>

<hint xmlns:pi="http://pretextbook.org/2020/pretext/internal">

  <p>

    <m>P_n</m>, the vector space of polynomials with degree at most <m>n</m>, has dimension <m>n+1</m> by <xref ref="theorem-exponent-laws"/>.  [Cross-reference is just a demo, content is not relevant.]  What happens if we relax the defintion and remove the parameter <m>n</m>?

  </p>

</hint>