Need help proving/disproving

I'm not sure how to best phrase this, so bear with me:
If you're given
abc = xyz
and
a+b+c = x+y+z
where a, b, c, x, y, z are all positive integers, is there a way to prove or disprove a = x, b = y, c = z?

I want to know if an integer shown in this way is unique.

>>16

Yes, just use polynomials.

in the case of just 3 variables. If they were not the same, up to re-ordering, then consider this polynomial.

f(t)=(t-a)(t-b)(t-c)= t^3 - (a+b+c)t^2 + (ab+ac+bc)t - abc
    =t^3 - (x+y+z)t^2 + (xy+xz+z)t - xyz [By the equalities]
    =(t-x)(t-y)(t-z)


so as long the base field K we're working is a UFD, this implies K[x] is also a UFD, and so we have (without loss of generality) a=x,b=y,c=z.