Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Imperative programming (resit) / 1 / d / Solution

Every positive divisor has a unique prime factorization $p_1^{e_1}\cdots p_k^{e_k}$ with $0\le e_j\le i_j$. Conversely every such exponent choice gives a divisor. The exponent choices are independent, so the product rule givesFor $n=1$ the factorization is empty, and the empty product is one, corresponding to the divisor one.