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Exercises 2.6 Sage 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.

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

2.

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

3.

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

4.

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

5.

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

6.

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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