Prove that two sequences of integers that have the same sum and product must be the same.

Question

Given two sequences of nondecreasing distinct positive integers such that $$x_1 + x_2 + ... + x_i = y_1 + y_2 + ... + y_i , i>0$$ and that $$x_1x_2 ... x_i = y_1y_2 ... y_i$$ Prove/disprove that the sequences are equal i.e. $$x_1 = y_1, x_2 = y_2, ... , x_i = y_i$$

I started with
Let $x_1x_2 ... x_i$ be $A$. If $A$ is prime, $x_1 = A = y_1$ (since $A$ cannot be factored any more) and we are done.

What I don't know is what happens when $A$ is not prime. Intuitively, it sounds true, and I cannot find any counter examples.

Answer 1

Counterexample:

$12+4+3 \ =\ 9+8+2$

$12\cdot4\cdot3 \ = \ 9\cdot8\cdot2$

Moreover, for $\ i>2\ ,\ $you can always find infinitely many counterexamples.

Answer 2

Let $x_1 = y_2, x_2 = y_1$ for a quick counter example.