Solution:

We know that $11$ is prime, which means that it only has $2$ divisors.

The prime factorization of $20$ is $$2^2 \cdot 5.$$ Recall that the number of divisors a number has is the product of all the exponents plus one in the prime factorization.

Here, that product would be $$(2 + 1) (1 + 1) = 3 \cdot 2 = 6.$$

Then $2 \cdot 6 = 12.$ We have the prime factorization of $12$ is $$2^2 \cdot 3.$$ This also has $$(2 + 1) (1 + 1) = 3 \cdot 2 = 6$$ divisors.

Thus, A is the correct answer.