 Multiplicative order

The multiplicative order of a unit $a$ modulo $N$ is the least positive $r$ for which $a^r\equiv 1(\mathrm{mod}N)$.

The multiplicative order of an integer \( a \) modulo \( n \) is defined as the smallest positive integer \( k \) such that \[ a^k \equiv 1 \mod n. \] In simpler terms, it is the smallest exponent \( k \) for which raising \( a \) to the power of \( k \) results in a value that, when divided by \( n \), leaves a remainder of 1.