Why does every number of shape ababab is divisible by $13$? Why does it seems like every number $ababab$, where $a$ and $b$ are integers $[0, 9]$ is divisible by $13$? 
Ex: $747474$, $101010$, $777777$, $989898$, etc...

Hint $\,\ {\rm mod}\ 13\!:\,\ 10\equiv 6^2\,\Rightarrow\, 10^6\equiv 6^{12}\equiv 1\,$ by little Fermat.
Hence $\, 0\equiv 10^6-1 \equiv (10^2\!-1)(10^4\!+10^2\!+1) \equiv 99\cdot 10101$
Thus $\,99\not\equiv 0\,\Rightarrow\,10101\equiv 0\,\Rightarrow\, ab\cdot 10101 = ababab\equiv 0,\,$ for $\, 0 \le ab \le 99$
