2.07:_Problem_Bank

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<section class="box-example"><span id="Problem_28"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 28</span></h2>

<ol>
    <li class="lt-math-10360">If you were counting in base four, what number would you say just before you said $100_{four}$?</li>
    <li class="lt-math-10360">What number is one more than $133_{four}$?</li>
    <li class="lt-math-10360">What is the greatest three-digit number that can be written in base four? What numbers come just before and just after that number?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_29"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 29</span></h2>

<p class="lt-math-10360">Explain what is wrong with writing $313_{two}$ or $28_{eight}$.</p>
</section>

<section class="box-example"><span id="Problem_30"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 30</span></h2>

<ol>
    <li class="lt-math-10360">Write out the base three numbers from $1_{three}$ to $200_{three}$.</li>
    <li class="lt-math-10360">Write out the base five numbers from $1_{five}$ to $100_{five}$.</li>
    <li class="lt-math-10360">Write the four base six numbers that come after $154_{six}$.</li>
</ol>
</section>

<section class="box-example"><span id="Problem_31"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 31</span></h2>

<p class="lt-math-10360">Convert each base ten number to a base four number. Explain how you did it.</p>

<p class="lt-math-10360">\[13, \qquad 8, \qquad 24, \qquad, 49 \nonumber \]</p>

<p class="lt-math-10360"><b>Challenges:</b></p>

<p class="lt-math-10360">\[0.125, \qquad 0.11111 \cdots = 0. \bar{1} \nonumber \]</p>
</section>

<section class="box-example"><span id="Problem_32"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 32</span></h2>

<p class="lt-math-10360">In order to use base sixteen, we need sixteen digits &mdash; they will represent the numbers zero through fifteen.  We can use our usual digits 0&ndash;9, but we need <i>new symbols</i> to represent the <i>digits</i> ten, eleven, twelve, thirteen, fourteen, and fifteen.  Here&rsquo;s one standard convention:</p>

<table class="mt-responsive-table">
    <tbody>
        <tr>
            <th class="mt-align-center"><strong>base ten</strong></th>
            <th class="mt-align-center"><strong>base sixteen</strong></th>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">7</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$7_{sixteen}$$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">8</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$8_{sixteen}$$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">9</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$9_{sixteen}$$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">10</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$A_{sixteen}$$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">11</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$B_{sixteen}$$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">12</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$C_{sixteen}$$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">13</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$D_{sixteen}$$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">14</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$E_{sixteen}$$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">15</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$F_{sixteen}$$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-10360" data-th="base ten">16</td>
            <td class="mt-align-center lt-math-10360" data-th="base sixteen">$$10_{sixteen}$$</td>
        </tr>
    </tbody>
</table>

<ol>
    <li class="lt-math-10360">Convert these numbers from base sixteen to base ten, and show your work:
    <p class="lt-math-10360">\[6D_{sixteen} \qquad AE_{sixteen} \qquad 9C_{sixteen} \qquad 2B_{sixteen} \nonumber \]</p>
    </li>
    <li class="lt-math-10360">Convert these numbers from base ten to base sixteen, and show your work:
    <p class="lt-math-10360">\[97 \qquad 144 \qquad 203 \qquad 890 \nonumber \]</p>
    </li>
</ol>
</section>

<section class="box-example"><span id="Problem_33"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 33</span></h2>

<p class="lt-math-10360">How many different symbols would you need for a base twenty-five system? Justify your answer.</p>
</section>

<section class="box-example"><span id="Problem_34"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 34</span></h2>

<p class="lt-math-10360">All of the following numbers are multiples of three.</p>

<p class="lt-math-10360">\[3, \quad 6, \quad 9, \quad 12, \quad 21,  \quad 27,  \quad 33,  \quad 60,  \quad 81,  \quad 99 \ldotp \nonumber \]</p>

<ol>
    <li class="lt-math-10360">Identify the <i>powers of </i>3 in the list. Justify your answer.</li>
    <li class="lt-math-10360">Write each of the numbers above in base three.</li>
    <li class="lt-math-10360">In base three: how can you recognize a <i>multiple of </i>3?  Explain your answer.</li>
    <li class="lt-math-10360">In base three: how can you recognize a <i>power of </i>3? Explain your answer.</li>
</ol>
</section>

<section class="box-example"><span id="Problem_35"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 35</span></h2>

<p class="lt-math-10360">All of the following numbers are multiples of five.</p>

<p class="lt-math-10360">\[5, \quad 10, \quad 15, \quad 25, \quad 55,  \quad 75,  \quad 100,  \quad 125,  \quad 625,  \quad 1000 \ldotp \nonumber \]</p>

<ol>
    <li class="lt-math-10360">Identify the <i>powers of </i>5 in the list. Justify your answer.</li>
    <li class="lt-math-10360">Write each of the numbers above in base five.</li>
    <li class="lt-math-10360">In base five: how can you recognize a <i>multiple of </i>5?  Explain your answer.</li>
    <li class="lt-math-10360">In base five: how can you recognize a <i>power of </i>5? Explain your answer.</li>
</ol>
</section>

<section class="box-example"><span id="Problem_36"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 36</span></h2>

<p class="lt-math-10360">Convert each number to the given base.</p>

<ol>
    <li class="lt-math-10360">$395_{ten}$ into base eight.</li>
    <li class="lt-math-10360">$52_{ten}$ into base two.</li>
    <li class="lt-math-10360">$743_{ten}$ into base five.</li>
</ol>
</section>

<section class="box-example"><span id="Problem_37"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 37</span></h2>

<p class="lt-math-10360">What bases makes theses equations true? Justify your answers.</p>

<ol>
    <li class="lt-math-10360">$$35 = 120 \_\_\_$$</li>
    <li class="lt-math-10360">$$41_{six} = 27 \_\_\_$$</li>
    <li class="lt-math-10360">$$52_{seven} = 34 \_\_\_$$</li>
</ol>
</section>

<section class="box-example"><span id="Problem_38"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 38</span></h2>

<p class="lt-math-10360">What bases makes theses equations true? Justify your answers.</p>

<ol>
    <li class="lt-math-10360">$$32 = 44\_\_\_$$</li>
    <li class="lt-math-10360">$$57_{eight} = 10 \_\_\_$$</li>
    <li class="lt-math-10360">$$31_{four} = 11 \_\_\_$$</li>
    <li class="lt-math-10360">$$15_{x} = 30_{y}$$</li>
</ol>
</section>

<section class="box-example"><span id="Problem_39"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 39</span></h2>

<ol>
    <li class="lt-math-10360">Find a base ten number that is twice the product of its two digits. Is there more than one answer? Justify what you say.</li>
    <li class="lt-math-10360">Can you solve this problem in any base other than ten?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_40"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 40</span></h2>

<ol>
    <li class="lt-math-10360">I have a four-digit number written in base ten. When I multiply my number by four, the digits get reversed.  Find the number.</li>
    <li class="lt-math-10360">Can you solve this problem in any base other than ten?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_41"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 41</span></h2>

<p class="lt-math-10360">Convert each base four number to a base ten number. Explain how you did it.</p>

<p class="lt-math-10360">\[13_{four} \quad 322_{four} \quad 101_{four} \quad 1300_{four} \nonumber \]</p>

<p class="lt-math-10360"><b>Challenges:</b></p>

<p class="lt-math-10360">\[0.2_{four} \qquad 0.111 \ldots_{four} = 0. \bar{1}_{four} \nonumber \]</p>
</section>

<section class="box-example"><span id="Problem_42"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 42</span></h2>

<p class="lt-math-10360">Consider this base ten number (I got this by writing the numbers from 1 to 60 in order next to one another):</p>

<p class="lt-math-10360">\[12345678910111213 \ldots 57585960 \ldotp \nonumber \]</p>

<ol>
    <li class="lt-math-10360">What is the largest number that can be produced by erasing one hundred digits of the number? (When you erase a digit it goes away. For example, if you start with the number 12345 and erase the middle digit, you produce the number 1245.) How do you <i>know</i> you got the largest possible number?</li>
    <li class="lt-math-10360">What is the smallest number that can be produced by erasing one hundred digits of the number? How do you <i>know</i> you got the smallest possible number?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_43"></span><h2 class="box-legend lt-math-10360"><span class="lt-icon-default">Problem 43</span></h2>

<p class="lt-math-10360">Can you find two different numbers (not necessarily single digits!) <img alt="a" height="8" src="http://pressbooks.oer.hawaii.edu/mathforelementaryteachers/wp-content/ql-cache/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.svg" title="Rendered by QuickLaTeX.com" width="9" /> and <img alt="b" height="13" src="http://pressbooks.oer.hawaii.edu/mathforelementaryteachers/wp-content/ql-cache/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.svg" title="Rendered by QuickLaTeX.com" width="8" /> so that $a_{b} = b_{a}$? Can you find more than one solution? Justify your answers.</p>
</section>

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