Exercise · ZW-UNIT-OF-A-RING-009 · ★★★ Advanced
For which of these values of \(n\) is every nonzero element of \(\mathbb{Z}_n\) a unit?
Choose one option.
Show answer
- D. 13
Explanation
A nonzero element \(a\) is a unit if \(\gcd(a,n)=1\); for this to hold for every \(a\) between \(1\) and \(n-1\), \(n\) must be prime. Of the given values only \(13\) is; in \(8,9,12\) there are elements such as \(2\), \(3\) and \(6\) sharing a factor with \(n\).
Review the concept: Unit of a ring
A unit of a ring \(R\) with identity \(1\) is an element \(u\) that has a multiplicative inverse \(v\) within \(R\).
\[uv=vu=1\]
More practice problems: unit of a ring
- ZW-UNIT-OF-A-RING-001 How many units does the ring ℤ_12 have?
- ZW-UNIT-OF-A-RING-002 Which of these elements is a unit of ℤ_10?
- ZW-UNIT-OF-A-RING-003 What is the multiplicative inverse of 5 in ℤ_17?
- ZW-UNIT-OF-A-RING-004 What are the units of the ring ℤ of integers?
- ZW-UNIT-OF-A-RING-005 How many units does the ring ℤ[i]=\a+bi: a,b∈ℤ\ have?
- ZW-UNIT-OF-A-RING-006 Which of these elements is a unit of ℤ[√2]?
- ZW-UNIT-OF-A-RING-007 If u and v are units of a commutative ring with identity, which of these elements is also a unit?
- ZW-UNIT-OF-A-RING-008 In a ring with 1ne0, can the element 0 be a unit?