5. Prime factorization Prove that every n ∈ N has a unique factorization

$$n = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k}$$,

(2.21)

where the p_i are prime numbers in increasing order and a_i ∈ N. (Hint: Use mathe-

matical induction on n. In the inductive step, argue by cases: what if n + 1 is prime;

what if it isn't?)