If $a,b,x$ and $y$ are integers greater than $1$ such that $a$ and $b$ have no common factors except $1$ and $x^a=y^b$, show that $x=n^b$ and $y=n^a$ for some integer $n$ greater than $1$.

Find all four-digit numbers having the following properties:

$\left(i\right)$ it is a square,

$\left(ii\right)$ its first two digits are equal to each other and

$\left(iii\right)$ its last two digits are equal to each other.