Solution
= Solution
By <unique prime factorization>, a positive <divisor> has precisely the form
$$
d=p_1^{e_1}\cdots p_k^{e_k},\qquad 0\leq e_j\leq i_j.
$$
There are $i_j+1$ independent choices for each <prime exponent>; different choices give different <divisors>, and every <divisor> occurs. The <multiplication principle> therefore gives \b[$\boxed{\tau(n)=\prod_{j=1}^{k}(i_j+1)}$]. For $n=1$, the empty <product> is $1$, giving its one positive <divisor>.