Question

The number of generators of the additive group \( \mathbb{Z}_{36} \) is equal to

• A. 6
• B. 12
• C. 18
• D. 36

Hint:

To find the number of generators of \( \mathbb{Z}_n \), use Euler's totient function \( \phi(n) \).

Answer 1

The Correct Option is B

Step 1: Understanding the question.
The group \( \mathbb{Z}_{36} \) is the additive group of integers modulo 36. We need to find the number of generators of this group, which are the elements whose orders are equal to 36. The order of an element \( x \) in \( \mathbb{Z}_n \) is the smallest integer \( k \) such that \( kx \equiv 0 \pmod{n} \).

Step 2: Finding the number of generators.
The number of generators of \( \mathbb{Z}_{n} \) is given by \( \phi(n) \), where \( \phi \) is the Euler's totient function. For \( n = 36 \), we compute \( \phi(36) \): \[ \phi(36) = 36 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right) = 36 \times \frac{1}{2} \times \frac{2}{3} = 12. \] Thus, the number of generators of \( \mathbb{Z}_{36} \) is 12.

Step 3: Conclusion.
The correct answer is (B) 12.