$"$ Show line by line calculation of Time Complexity".$($a$)$ Derive the best$-$case, and the worst$-$case running$-$time equations for the following

function favouriteSum and represent using Asymptotic Notation.

bool favouriteSum$($int n$,$ int m$)\{$

int sum $=0$;

for$($int i$=1$;in;i$++)\{$

for$($int j$=1$;ji;j$++)\{$

sum $-$ sum $+($i$+$j$)$;

$)$

for$($int i$=2$;i$*$im;i$++)\{$

if$($m$\%$i$==0)$

return false;

if $($sumsm$==0)$

return true;

else

return false;

$($b$)$ Derive the exact$-$cost equation for the running$-$time of the following function and

show that the time complexity is $O(nlo{g}_{4}n).$

Fvoid Function$($int n$)($

int prod $=1$;

for$($int i$=$n; i$>=1$; i$--)\{$

for$($int j$=$n; j$>=1$; j$=$j$/4)\{$

prod $=$ prod $*($i$+$j$)$;

$1$

$\}$

printf$("\%$d

$",$ prod$)$;