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...

Note that
$$
[ababab] = a\times 101010 + b \times 10101 =
13 (7770a + 777b)
$$

Also noteworthy:
$$
10101 = 1 + 10^2 + 10^4 \equiv \\
1 + 3^2 + 3^4 = 1 + 9 + 9^2 \equiv\\
1 +(-4) + (-4)^2 = 1 - 4 + 16 = 13 \equiv 0
$$
where $\equiv$ indicates equivalence modulo $13$.
