The Java method below calculates the greatest common divisor of two integers $($num $1,$ num$2).$

The valid input range of num $1$ and num $2$ is specified as follows:

$0\backslash$leqnum$1\backslash$leq$100000\backslash$leqnum$2\backslash$leq$5000$

public static int gcd$($int num$1,$ int num$2)\{$

int a $=$ num$1$;

int b $=$ num$2$;

if $($a$<mn>0</mn><mtext></mtext><mi>|</mi><mi>|</mi>$ b$<mn>0</mn><mtext></mtext><mi>|</mi><mi>|</mi>$ a$>10000||$ b $>5000)$

l

System.out.println$("$Invalid input"$)$;

return $-1$;

$\}$ else $\{$

int i $=0$;

while $($a $!=0)\{$

$++$i;

int temp $=$ a;

a $=$ b $\%$ a;

b $=$ temp;v

$\}$

System.out.println$("$GCD Calculation required $"+$

i $+"$ iterations"$)$;

return b;

$\}$

a$)$ Draw the corresponding Control Flow Graph and label the nodes of the CFG with the respective line numbers in the source code. $[7$ points$]$

b$)$ Draw the prime decomposition tree. $[4$ points$]$

c$)$ How many test cases are needed $($at a minimum$)$ to achieve the following coverage criteria? Please list the identified test cases $[4$ points$]$

Simple path coverage

Basis path coverage

d$)$ Provide provide all $($partial$)$ paths to achieve all DU$-$path coverage with respect to variable'a$.$ All paths should be defined in terms of nodes traversed in the CFG$.[4$ points$]$

e$)$ Create the following backward slice $ $($fb$\},16).$ Simply list the line numbers of the statementsto be included in the slice. $[4$ points$]$