Take $X = 10, Y = 3$
In that case,
$ \underline{\text{Before Iteration 1:}}$
$res = 1 , a = 10 , b = 3$
All options are satisfied here.
$ \underline{\text{ Iteration 1:}}$
- $while(b!=0) \\ \implies 3!=0 \hspace{0.8cm} \checkmark$
- $if(b\%2==0) \\ \implies 3\%2==0 \hspace{0.8cm} \times$
- $else$
$\quad res = res*a \\ \quad \implies res= 1*10 =10 $
$b =b-1 \\ \implies b= 3-1 =2 $
$ \underline{\text{Before Iteration 2:}}$
$res = 10 , a = 10 , b = 2$
option A : $X^Y = a^b \\ \implies 10^3 = 10^2 \hspace{0.8cm} \times$
option B : $(res * a)^Y = (res * X)^b \\ \implies (10*10)^3 = (10*10)^2 \hspace{0.8cm} \times$
option C : $X^Y = res * a^b \\ \implies 10^3 = 10 * 10^2 \hspace{0.8cm} \checkmark$
option D : $X^Y = (res*a)^b \\ \implies 10^3 = (10*10)^2 \hspace{0.8cm} \times$
Lets see one more iteration to verify option C.
$ \underline{\text{ Iteration 2:}}$
$res = 10 , a = 10 , b = 2$
- $while(b!=0) \\ \implies 2!=0 \hspace{0.8cm} \checkmark$
- $if(b\%2==0) \\ \implies 2\%2==0 \hspace{0.8cm} \checkmark$
$a= a*a$
$\quad = 10*10=100$
$b= \dfrac{b}{2}$
$\quad = \dfrac{2}{2}=1$
$ \underline{\text{Before Iteration 3:}}$
$res = 10 , a = 100 , b = 1$
Option C : $X^Y = res * a^b \\ \implies 10^3 = 10 * 100^1 \\ = 10^3 \hspace{0.8cm} \checkmark$
Option C is answer