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The MultiplicativeOrder function computes the multiplicative order of m modulo n, which is defined as the least positive integer exponent i such that m^i is congruent to $1$ modulo n.

Alternatively, the multiplicative order can be defined as the order of the cyclic group generated by m under multiplication modulo n.

The multiplicative order exists if and only if m and n are coprime. In the case that it does not exist, an error message is displayed.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\mathrm{MultiplicativeOrder}\left(7\,18\right)$

$3$

$\mathrm{seq}\left(7^i\mathbf{mod}18\,i=1..3\right)$

$7,13,1$

$\mathrm{Totient}\left(18\right)$

$6$

$\mathrm{MultiplicativeOrder}\left(11\,18\right)$

$6$

$\mathrm{PrimitiveRoot}\left(18\,\mathrm{greaterthan}=10\right)$

$11$

$\mathrm{MultiplicativeOrder}\left(5\,25\right)$

Error, (in NumberTheory:-MultiplicativeOrder) the arguments 5 and 25 are not coprime

The NumberTheory[MultiplicativeOrder] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016 .