Multiplicativity of the divisor function

Statement

Let $k$ be a natural number. The divisor function $\sigma_k$ is a multiplicative function. That is, $\sigma_k(mn) = \sigma_k(m) \sigma_k(n)$ for all coprime positive integers $m$ and $n$.

Remarks

The proof: for coprime $m$ and $n$, each divisor of $mn$ is a unique product $d e$ with $d \mid m$ and $e \mid n$. The function $\sigma_k$ is not completely multiplicative: $\sigma_0(4) = 3$, but $\sigma_0(2)^2 = 4$. At a prime power, $\sigma_k(p^e) = 1 + p^k + p^{2k} + \dots + p^{ek}$. Multiplicativity then gives $\sigma_k(n)$ for all $n$.

Explanations

L1 human

For example, $\sigma(4) = 7$ and $\sigma(3) = 4$, so $\sigma(12) = 28$. This works because $4$ and $3$ share no prime factor. Each divisor of $12$ is a divisor of $4$ times a divisor of $3$.