$[$Utilities$]$ Implement the methods in Primes.java.

import java.util.$*$;

import java.io$.*$;

class Primes

$\{$

$/*$ in this assignment, we will only work with the first $1000$ primes $*/$

static int NUM$\_$PRIMES $=1000$;

$/*$ the square of the value of the largest prime number used in this

assignment $*/$

static int SKY $=-1$;

$/*$ the values of the first NUM$\_$PRIMES prime numbers $*/$

static int$[]$ primes $=$ new int$[$NUM$\_$PRIMES$]$;

$/*$ load the first NUM$\_$PRIMES prime numbers from the file "primes.txt$"$

$*$ into the static array 'primes'. This method must also

$*$ initialize the $\text{'}$SKY$\text{'}$ static variable.

$*/$

public static void loadPrimes$()$

$\{$

$/*$ To be completed $*/$

$\}//$ loadPrimes method

$/*$ return true if and only if its input is one of the first NUM$\_$PRIMES

$*$ prime numbers.

$*/$

public static boolean isPrime$($int n$)$

$\{$

$/*$ To be completed $*/$

return false; $/*$ Delete this line in your submission $*/$

$\}//$ isPrime method

$/*$ given an integer, return a $2-$element array containing its two

$*$ prime factors if the input is the product of two primes $($with

$*$ the smallest prime in the first array location$)$; otherwise

$*$ return an empty $(0-$element$)$ array.

$*/$

public static int$[]$ factor$($int n$)$

$\{$

$/*$ To be completed $*/$

return null; $/*$ Delete this line in your submission $*/$

$\}//$ factor method

$/*$ return the greatest common divisor of the two input integers,

$*$ which are assumed to be positive. The implementation of this

$*$ method must be iterative. No recursion allowed! You are also

$*$ not allowed to use any $\text{'}$gcd$\text{'}$ method in any of the Java API

$*$ classes. You may not use any helper methods. You must implement

$*$ this method from first principles.

$*/$

public static int gcd$($int m$,$int n$)$

$\{$

$/*$ To be completed $*/$

return $-1$; $/*$ Delete this line in your submission $*/$

$\}//$ gcd method

$/*$ return true if and only if alpha is a primitive root modulo q

$*/$

public static boolean isPrimitiveRoot$($int alpha, int q$)$

$\{$

$/*$ To be completed $*/$

return false; $/*$ Delete this line in your submission $*/$

$\}//$ isPrimitiveRoot method

$/*$ return the largest primitive root of q $($or $-1,$ if q does not have

$*$ a primitive root$).$

$*/$

public static int pickAlpha$($int q$)\{$

$/*$ To be completed $*/$

return $-1$; $/*$ Delete this line in your submission $*/$

$\}//$ pickAlpha method

$/*$ This method is used for testing.

$*$ Do NOT modify it

$*/$

public static void main$($String$[]$ args$)$

$\{$

if $($args$.$length $==1)$

$\{$

loadPrimes$()$;

if $($args$[0].$equals$("$load$"))$

$\{$

System.out.println$("$Testing loadPrimes$()")$;

loadPrimes$()$;

System.out.println$($primes$[0])$;

System.out.println$($primes$[$NUM$\_$PRIMES$-1])$;

System.out.println$($SKY$)$;

$\}$ else if $($args$[0].$equals$("$isPrime$"))$

$\{$

for$($int i $=0$; i $<20$; i$++)$

$\{$

System.out.println$($isPrime$($i$))$;

$\}$

$\}$ else if $($args$[0].$equals$("$factor$"))$

$\{$

for$($int i $=0$; i $<1000$; i$++)$

$\{$

int$[]$ result $=$ factor$($i$)$;

if $($result$.$length $==2)$

$\{$

System.out.format$("\%$d is the product of $\%$d and $\%$d

$",$

i$,$ result$[0],$ result$[1])$;

$\}$

$\}$

$\}$ else if $($args$[0].$equals$("$gcd$"))$

$\{$

for$($int i $=1$; i $<300$; i $+=3)$

$\{$

for$($int j $=5$; j $<200$; j $+=13)$

$\{$

System.out.println$("$gcd$("+$ i $+","+$ j $+")="+$

gcd$($i$,$j$))$;

$\}$

$\}$

$\}$ else if $($args$[0].$equals$("$primRoot$"))$

$\{$

for$($int i $=0$; i $<20$; i$++)$

$\{$

int n $=$ primes$[$i$]$;

System.out.print$($n $+"$: $")$;

for$($int j $=1$; j $<$ n; j$++)$

$\{$

if $($isPrimitiveRoot$($j$,$n$))$

$\{$

System.out.print$($ j $+"")$;

$\}$

$\}$

System.out.println$()$;

$\}$

$\}$ else if $($args$[0].$equals$("$pickAlpha$"))$

$\{$

for$($int i $=0$; i $<20$; i$++)$

$\{$

int q $=$ primes$[$i$]$;

System.out.println$($q $+"$: $"+$ pickAlpha$($q$))$;

$\}$

$\}$

$\}$

$\}$

$\}//$ Primes class