Exercise · ZW-MULTIPLICATIVE-ORDER-004 · ★★☆ Intermediate
If \( \operatorname{ord}_n(a)=k \) and \( a^m \equiv 1 \pmod{n} \), what relation must hold?
Choose one option.
Show answer
- B. \( k \mid m \)
Explanation
Since \( a^m \equiv 1 \) holds exactly when \( k \mid m \), the order controls all such exponents. The other relations fail: for example \( m=2k \) satisfies the condition but \( m \neq k \).
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-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 ?
- ZW-MULTIPLICATIVE-ORDER-009 It is claimed that ord_7(a)=ord_7(a^2) for every invertible a . Which value of a disproves the clai…