<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>
Exercises 2.6 Sage Exercises
View Source for 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.
View Source for 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>
Verify that \(1\) is the greatest common divisor of your two primes from the previous exercises.
4.
View Source for 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>
Find two integers that make a “linear combination” of your two primes equal to \(1\text{.}\) Include a verification of your result.
5.
View Source for exercise
<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 \(d=11\text{.}\)