Yes, formulating P(k) is the key in constructing a proof by induction.

Here, in fact, you have a standard case. If your theorem can be formulated as, "For all integers n >= n0, P(n)", then usually you can take P(n) as the induction statement. Here the theorem is, "For all integers n >= 1, for all primes p and sequences of integers a(1), ..., a(n), if p | a(1) * ... * a(n), then p | a(i) for some i". So, just strip the universal quantification over n and you get your P(n).

The base case for n = 1 is obvious, and for the induction step you need to use

Euclid's lemma.