Exercises 2.6 Sage Exercises

    <exercises xml:id="integers-sage-exercises">
    <title>Sage Exercises</title>

    <introduction>
        <p>These exercises are about investigating basic properties of the integers, something we will frequently do when investigating groups.  Use the editing capabilities of a Sage worksheet to annotate and explain your work.</p>
    </introduction>

    <exercise number="1">
        <statement>
            <p>Use the <c>next_prime()</c> command to construct two different 8-digit prime numbers and save them in variables named <c>a</c> and <c>b</c>.</p>
        </statement>
    </exercise>

    <exercise number="2">
        <statement>
            <p>Use the <c>.is_prime()</c> method to verify that your primes <c>a</c> and <c>b</c> are really prime.</p>
        </statement>
    </exercise>

    <exercise number="3">
        <statement>
            <p>Verify that <m>1</m> is the greatest common divisor of your two primes from the previous exercises.</p>
        </statement>
    </exercise>

    <exercise number="4">
        <statement>
            <p>Find two integers that make a <q>linear combination</q> of your two primes equal to <m>1</m>.  Include a verification of your result.</p>
        </statement>
    </exercise>


    <exercise number="5">
        <statement>
            <p>Determine a factorization into powers of primes for <m>c=4\,598\,037\,234</m>.</p>
        </statement>
    </exercise>

    <exercise number="6">
        <statement>
            <p>Write a compute cell that defines the same value of <c>c</c> again, and then defines a candidate divisor of <c>c</c> named <c>d</c>.  The third line of the cell should return <c>True</c> if and only if <c>d</c> is a divisor of <c>c</c>.  Illustrate the use of your cell by testing your code with <m>d=7</m> and in a new copy of the cell, testing your code with <m>d=11</m>.</p>
        </statement>
    </exercise>

</exercises>