## 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 \(d=11\text{.}\)