Discussion
How many units are there in \(\mathbf{Z}_{12}\) ?
Answer: B
\(\mathbf{Z}_{12}\) has 4 units.
A class a is a unit mod 12 exactly when \(\gcd(a, 12) = 1\). Those classes are 1, 5, 7 and 11, so there are \(\varphi(12) = 12\left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right) = 4\) units. Each is its own inverse: \(5^{2} = 25 \equiv 1\), \(7^{2} = 49 \equiv 1\), \(11^{2} = 121 \equiv 1\).
3 and 5 miscount the coprime residues, and 2 counts only \(\pm 1\).
No comment is present. Be the first to comment.
Loading…
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development