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BUILT-IN MATHEMATICA SYMBOL

MultiplicativeOrder

gives the multiplicative order of k modulo n, defined as the smallest integer

such that

gives the generalized multiplicative order of k modulo n, defined as the smallest integer

such that

for some

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

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

The multiplicative order of 7 modulo 108:

Out[1]=

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

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

repeat with period

The repetition period in Rule 90 for odd n divides

EulerPhi is a multiple of the multiplicative order:

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