Exercise · ZW-MULTIPLICATIVE-ORDER-009 · ★★★ Advanced
It is claimed that \( \operatorname{ord}_7(a)=\operatorname{ord}_7(a^2) \) for every invertible \( a \). Which value of \( a \) disproves the claim?
Choose one option.
Show answer
- A. \( a=3 \)
Explanation
With \( a=3 \) the order is 6, while \( 3^2 \equiv 2 \pmod 7 \) has order 3. The other values satisfy the equality, so they do not disprove the claim.
Review the concept: Multiplicative order
The multiplicative order of \( a \) modulo \( n \), \( \operatorname{ord}_n(a) \), is the smallest positive exponent that returns \( a \) to the unit class modulo \( n \).
\[\operatorname{ord}_n(a)=\min\{k\ge 1: a^k\equiv 1 \pmod{n}\}\]
More practice problems: multiplicative order
- ZW-MULTIPLICATIVE-ORDER-001 What is the definition of the multiplicative order of a modulo n ?
- ZW-MULTIPLICATIVE-ORDER-002 What is ord_5(4) ?
- ZW-MULTIPLICATIVE-ORDER-003 What condition must a satisfy for ord_n(a) to exist?
- ZW-MULTIPLICATIVE-ORDER-004 If ord_n(a)=k and a^m ≡ 1 pmodn , what relation must hold?
- ZW-MULTIPLICATIVE-ORDER-005 What is ord_10(3) ?
- ZW-MULTIPLICATIVE-ORDER-006 Compute ord_9(2) .
- ZW-MULTIPLICATIVE-ORDER-007 If ord_13(a)=6 , what is ord_13(a^2) ?
- ZW-MULTIPLICATIVE-ORDER-008 How is ord_n(a^-1) related to ord_n(a) , when a is invertible modulo n ?