$4^{51}+4^{52}+4^{53}+4^{54}+4^{55}$ is divisible by:

Factor out the lowest power term $4^{51}$:
$$4^{51}(1+4^1+4^2+4^3+4^4)$$
$$=4^{51}(1+4+16+64+256)$$
$$=4^{51}(341)$$
Now check the options to see which number divides 341 perfectly:
$$\frac{341}{11}=31$$
Since 341 is a multiple of 11, the entire expression is completely divisible by 11.