Explanation:

To find the total number of generators of the cyclic group Z12 under addition modulo 21, we need to determine the number of elements in the group that generate the entire group.

Description of Cyclic Group:

A cyclic group is a group that is generated by a single element. In other words, every element in the group can be expressed as a power of the generator. In this case, the generator is an integer modulo 21.

Counting the Generators:

To count the number of generators of the cyclic group Z12, we need to find the number of elements in the group that generate the entire group.

We know that the order of the group Z12 is 12, which means it contains 12 elements. To find the generators, we need to determine which elements generate the entire group when raised to different powers.

Since 21 is a prime number, the group Z12 is isomorphic to the group Z21. In Z21, the generators are the integers relatively prime to 21. Therefore, to find the generators of Z12, we need to find the integers relatively prime to 21.

Finding the Integers Relatively Prime to 21:

The integers relatively prime to 21 are those integers that do not have any common factors (other than 1) with 21. These integers are 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, and 20.

Counting the Generators:

Out of the 12 elements in Z12, 12 of them are relatively prime to 21. Therefore, the total number of generators of the cyclic group Z12 under addition modulo 21 is 12.

Correct Answer:
The correct answer is option 'A', 18.