> [!NOTE] Definition > A multiplicative function is an [[Arithmetic Function|arithmetic function]], $f,$ such that for all $a,b\in \mathbb{N}^{+}$ that [[Coprime Integers|coprime]], $f(ab)=f(a)f(b)$ An arithmetic function _f_(_n_) is said to be **[[Totally Multiplicative Functions|totally multiplicative]]** if $f(ab)=f(a)f(b)$ holds _for all_ positive integers _a_ and _b_, even when they are not coprime. # Applications ###### Examples: - [[Identity function is the only strictly increasing multiplicative function with f(2)=2]].