OK, here is a sketch. Suppose that p, q, r are prime, i, j₁, j₂ and k are positive integers, and

,

. In the general case, the variables here should be vectors. Without loss of generality, let

. Then

.

Let

. Then the only primes that could divide s are p, q and r. If p | s, then

, a contradiction. Similarly, r | s is false. Thus

for some j. Moreover,

; otherwise q ∈ GCD(x, y, z). Finally, if

, then q | s implies

, a contradiction. Therefore,

,

and

.

See if you can shorten this and if it can be extended to the general case.