###### 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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\newcommand{\notdivide}{\nmid}
\newcommand{\notsubset}{\not\subset}
\newcommand{\lcm}{\operatorname{lcm}}
\newcommand{\gf}{\operatorname{GF}}
\newcommand{\inn}{\operatorname{Inn}}
\newcommand{\aut}{\operatorname{Aut}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\cis}{\operatorname{cis}}
\newcommand{\chr}{\operatorname{char}}
\newcommand{\Null}{\operatorname{Null}}
\newcommand{\transpose}{\text{t}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\)

These exercises are about investigating basic properties of the integers, something we will frequently do when investigating groups. Sage worksheets have extensive capabilities for making new cells with carefully formatted text, include support for LaTeX syntax to express mathematics. So when a question asks for explanation or commentary, make a new cell and communicate clearly with your audience.

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