Solution
= Solution
Every positive <divisor> has a unique <prime factorization> $p_1^{e_1}\cdots p_k^{e_k}$ with $0\leq e_j\leq i_j$. Conversely every such exponent choice gives a <divisor>. The exponent choices are independent, so the <product rule> gives
$$
\boxed{\tau(n)=\prod_{j=1}^{k}(i_j+1).}
$$
For $n=1$ the factorization is empty, and the <empty product> is one, corresponding to the <divisor> one.