The multiplicative order measures the period of powers of a modulo n. Euler theorem guarantees that the order divides phi(n) whenever a and n are coprime.
⚠The calculator tests divisors of phi(n). Inputs should be ordinary browser-size integers because factorization of large n can be slow.
What Is Multiplicative Order?
Multiplicative order is the length of the repeating cycle generated by powers of a modulo n. It connects modular arithmetic to group theory and primitive roots.
Coprime Required
The order exists only for units modulo n.
Smallest Exponent
The answer is the first k where a^k returns to 1.
Divides phi(n)
Euler theorem limits the possible k values.
Cycle Length
The order is the period of repeated multiplication by a.
Example: For a = 2 and n = 9, 2^1 = 2, 2^2 = 4, 2^3 = 8, and 2^6 = 1 mod 9, so ord_9(2) = 6.
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