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.
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