SB Sage Exercises

Exercises 2.6 Sage Exercises

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<exercises xml:id="integers-sage-exercises" xml:base="sage/integers-sage-exercises.xml">
  <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>

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.

1.

View Source for exercise

<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>

Use the next_prime() command to construct two different 8-digit prime numbers and save them in variables named a and b.

2.

View Source for 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>

Use the .is_prime() method to verify that your primes a and b are really prime.

3.

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<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>

Verify that \(1\) is the greatest common divisor of your two primes from the previous exercises.

4.

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<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>

Find two integers that make a “linear combination” of your two primes equal to \(1\text{.}\) Include a verification of your result.

5.

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

Determine a factorization into powers of primes for \(c=4\,598\,037\,234\text{.}\)

6.

View Source for 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>

Write a compute cell that defines the same value of c again, and then defines a candidate divisor of c named d. The third line of the cell should return True if and only if d is a divisor of c. Illustrate the use of your cell by testing your code with \(d=7\) and in a new copy of the cell, testing your code with \(d=11\text{.}\)

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