Modular multiplicative inverse ($a^{-1}(\mathrm{mod}n)$)

The modular multiplicative inverse of an integer \( a \) with respect to a modulus \( m \) is another integer \( x \) such that the product \( ax \equiv 1 \mod m \). In other words, when \( a \) is multiplied by \( x \) and then divided by \( m \), the remainder is 1.