Find the remainder!

Let $x = 2^{133}$ . Note that $133 = 7 * 19$

$x = 2^{133} \equiv 2^{133 \pmod{\phi(7)}} \pmod{7}$

$\rightarrow x \equiv 2^{133 \pmod{6}} \equiv 2^1 \equiv 2 \pmod{7}$

And, $x = 2^{133} \equiv 2^{133 \pmod{\phi(19)}} \pmod{19}$

$\rightarrow x \equiv 2^{133 \pmod{18}} \equiv 2^{7} \equiv 128 \equiv 14 \pmod{19}$

So we have $x \equiv 2 \pmod{7} , x \equiv 14 \pmod{19}$

We can write $x = 7y + 2$

$\rightarrow x= 7y + 2 \equiv 14 \pmod{19}$

$7y \equiv 12 \pmod{19}$

$77y \equiv 132 \pmod{19}$

$y \equiv 18 \pmod{19}$

We can write this as $y = 19z + 18$

$x = 7y + 2 = 7(19z + 18) + 2 = 133z + 126 + 2 = 133z + 128$

$\rightarrow x \equiv \boxed{128} \pmod{133}$