To determine the number of generators of the additive group Z36, we need to understand the properties of generators and the structure of Z36.

Generators:
In group theory, a generator is an element that, when combined with itself or other elements of the group, can generate all the elements of the group. In other words, a generator can produce all the elements of a group by repeated addition.

The Structure of Z36:
Z36 is the set of integers modulo 36, or the integers from 0 to 35. Addition in Z36 is performed modulo 36, meaning that if the sum exceeds 35, it wraps around to the beginning of the set.

Finding Generators:
To determine the number of generators of Z36, we can start by considering the possible generators of Z36. A generator must be an element that has no smaller positive integer as its multiple. In other words, if we take a generator as 'g', it should not be possible to express 'g' as 'n*g' for any positive integer 'n' less than 'g'.

To find the generators of Z36, we can start by selecting any element 'g' from Z36 and check if it generates all the elements of Z36. We can do this by repeatedly adding 'g' to itself until we get back to the original element. If this process generates all the elements of Z36, then 'g' is a generator.

Calculating the Generators:
Let's consider an example to find the generators of Z36.

Starting with the element 1, we can add it to itself repeatedly:

1 + 1 = 2
2 + 1 = 3
...
35 + 1 = 0 (wrapping around to the beginning)

By adding 1 repeatedly, we generate all the elements of Z36. Therefore, 1 is a generator of Z36.

Now, let's repeat the process with the element 2:

2 + 2 = 4
4 + 2 = 6
...
34 + 2 = 0 (wrapping around to the beginning)

By adding 2 repeatedly, we also generate all the elements of Z36. Therefore, 2 is another generator of Z36.

We can continue this process with other elements of Z36, and it turns out that the possible generators of Z36 are 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35. There are 12 elements that generate all the elements of Z36.

Therefore, the correct answer is option 'B': 12.