###### 1

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

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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.

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

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

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

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

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

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