Is knowing the Sum and Product of k different natural numbers enough to find them?

Question

Can we uniquely identify the set of k different natural numbers (no two are the same) by knowing only their sum and product (and the value of k itself)?

Answer 1

{4,9,10} and {5,6,12} have the same sum (23) and product (360) so the answer is no.

As such then {4k,9k,10k} and {5k,6k,12k} also have the same sum and product so there is a multitude of counter examples. I'm imagine there are others based on different triples as well.

Answer 2

I think the minimal counterexample (in the sense of minimal sum) are the triples $\{2,8,9\}$ and $\{ 3,4,12 \}$ for which we have $$ 2+8+9 = 19 \qquad 2\cdot 8\cdot 9 = 144$$ and $$ 3 + 4 + 12 = 19 \qquad 3 \cdot 4 \cdot 12 = 144.$$