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Built-in Mathematica Symbol

MultiplicativeOrder

MultiplicativeOrder[k, n]
gives the multiplicative order of k modulo n, defined as the smallest integer m such that

MultiplicativeOrder[k, n, {r₁, r₂, ...}]
gives the generalized multiplicative order of k modulo n, defined as the smallest integer m such that

for some i.

Integer mathematical function, suitable for both symbolic and numerical manipulation.

MultiplicativeOrder returns unevaluated if there is no integer m satisfying the necessary conditions.

The multiplicative order of 7 modulo 108:

Evaluate for large arguments:

MultiplicativeOrder works for negative first arguments:

Generalized multiplicative order:

Find all primitive roots modulo 43:

Find a solution to Mod[5^k, 7]=2:

The function digitCycleLength gives the digit period for any rational number r in base b:

This shows that the decimal representation of

in base

repeats every

digits.

The digits of

in base 4 repeat with period 6:

The repetition period in Rule 90 for odd n divides q[n]:

EulerPhi is a multiple of the multiplicative order:

The multiplicative order of a primitive root modulo n is EulerPhi[n]: