Show that $xyxyxy$ is not a perfect power. If $N=xyxyxy$ where $x$ and $y$ are digits.
Show that $N$ cannot be a perfect power, i.e. $N\ne a^b$, where $a$ and $b$ are positive integers and $b>1$.
My work
$xy|xyxyxy$
and $\frac{xyxyxy}{xy}=10101$
After factorizing of $10101$ we will get other factors of $xyxyxy$
Factors of $xyxyxy$ : 1, $xy$, 3, 7, 13, 21, 37, 39, 91, 111, 259, 273, 481, 777, 1443, 3367, 10101, $xyxyxy$
If I have missed any factor then please include that.
This is all what I have done, please help me.  

If $xyxyxy$ is a $k$-th power, then all primes dividing it must occur at least $k$ times in the factorization.  $10101=3\cdot7\cdot13\cdot37$; so if $10101$ divides $xyxyxy$, you need at least $k-1$ more $3$, $7$, $13$ and $37$'s in the factorization of $xyxyxy/10101 = xy$, which is impossible since $xy$ only has two digits.
