 # PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

## Exercises2.6Sage 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 $$d=11\text{.}$$