Exercise · ZW-INVERSE-ELEMENT-011 · ★★★ Advanced
In \(\mathbb{Z}_7\setminus\{0\}\) under multiplication, which \(x\) satisfies \(3\cdot x\cdot 3=1\)?
Choose one option.
Show answer
- B. 4
Explanation
Since \(3\cdot 3=9\equiv 2\pmod 7\), the equation becomes \(2x\equiv 1\); since \(2\cdot 4=8\equiv 1\), \(x=4\). Check: \(3\cdot 4\cdot 3=36\equiv 1\pmod 7\).
Review the concept: Inverse element
Given \(a\in G\), an element \(b\in G\) is an inverse of \(a\) if operating in both orders gives the neutral element.
\[a\ast a^{-1}=a^{-1}\ast a=e\]
More practice problems: inverse element
- ZW-INVERSE-ELEMENT-001 What is the inverse of 5 in (ℤ, +)?
- ZW-INVERSE-ELEMENT-002 What is the inverse of 3/4 under multiplication?
- ZW-INVERSE-ELEMENT-003 In a group, the inverse of an element is...
- ZW-INVERSE-ELEMENT-004 Which identity is correct?
- ZW-INVERSE-ELEMENT-005 If a⁻¹ is the inverse of a, then...
- ZW-INVERSE-ELEMENT-006 The additive inverse of -5 is...
- ZW-INVERSE-ELEMENT-007 Which statement correctly describes the inverse of an element of a group?
- ZW-INVERSE-ELEMENT-008 Which condition guarantees that a square matrix has an inverse?