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 $1$. Include a verification of your result.

##### 5

Determine a factorization into powers of primes for $c=4\,598\,037\,234$.

##### 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 $d=11$.