There is no power of 2 which is divisible by 11

By Fermat's Little Theorem, we have $a^{p-1}\equiv1\pmod{p}$ for prime $p$ and integer $a$ such that $\gcd{(a,p)}=1$ .

Then, $2^{10}\equiv1\pmod{11}$

Raise both sides to the power of $202$ :

$2^{2020}\equiv1\pmod{11}$

Multiply both sides by $2^2=4$ :

$2^{2022}\equiv4\pmod{11}$

Add $7$ to both sides:

$2^{2022}+7\equiv11\equiv0\pmod{11}$

Thus, $11$ divides $2^{2022}+7$ .

Try $\text{1}$ : $\text{2}^{18}={262144+7=262151}$

$\text{262151 not divisible by 11}$

Try $\text{2}$ :

$\text{2}^{20}={2048576+7=2048583}$

$\text{2048583 not divisible by 11}$

Try $\text{3}$

$\text{2}^{16}={65536+7=65543}$

$\text{65543 not divisible by 11}$

Try $\text{4}$ :

$\text{2}^{22}={4194304+7=4194311}$

$\text{4194311 divisible by 11}$

Hence the answer is $\large{2}^{2022}+7$ .