Number of Generators of Group Z16

Z16 is the cyclic group of integers modulo 16, which means it consists of the elements {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. To find the number of generators of Z16, we need to understand the concept of generators and calculate the value of φ(16).

Definition of Generators:
In group theory, a generator of a group G is an element that, when raised to the powers of all positive integers, produces all the elements of the group. In other words, a generator is an element that generates the entire group through its powers.

Calculating φ(16):
φ(16) represents Euler's totient function, which gives the number of positive integers less than or equal to n that are coprime to n. In this case, we want to calculate φ(16). To do this, we need to find the number of positive integers less than or equal to 16 that are coprime to 16.

Since 16 is not a prime number, we can use the formula φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pn), where p1, p2, ..., pn are the prime factors of n.

The prime factors of 16 are 2 and 2. Applying the formula, we get:
φ(16) = 16 * (1 - 1/2) * (1 - 1/2) = 16 * (1/2) * (1/2) = 8.

Finding the Number of Generators:
The number of generators of a cyclic group is equal to the number of positive integers less than or equal to n that are coprime to n. In this case, n = 16.

Since φ(16) = 8, the number of generators of Z16 is 8.

Therefore, the correct answer is option 'B', which states that there are 8 generators in the group Z16.