A multiplicative function is a special type of arithmetic function defined on the positive integers. It has the property that if two numbers are coprime (they share no common factors other than 1), the function's value at their product equals the product of their individual function values. In mathematical terms, if f is a multiplicative function and a and b are coprime, then f(ab) = f(a) \cdot f(b) . Common examples of multiplicative functions include the Euler's totient function \phi(n) and the divisor function d(n) . These functions play a significant role in number theory, particularly in the study of prime numbers and their distributions. Understanding multiplicative functions helps in analyzing the properties of integers and their relationships.