<s>[INST] $3$ numbers $x,y,z$ pairwise relatively prime, sum of any two is multiple of third, find $abc$ 
Three distinct positive integers $x,y,z$ are pairwise relatively prime, and the sum of any two is a multiple of the third one. Find $xyz$.

I don't know how to start.
 [/INST] Suppose WLOG $x < y < z$. Then $z \mid x + y < 2 z$, and there are no other multiples of $z$ between $z$ and $2 z$. It follows that $x + y = z$.
Now $x \mid y + z = x + 2 y$, so $x \mid 2 y$, and since $\gcd(x, y) = 1$, we have $x \mid 2$, and thus $x = 1$ or $x = 2$.
Similarly $y \mid x + z = 2 x + y$, so as above $y = 1$ or $y = 2$. Since $x < y$, we must have  $x = 1$, $y = 2$, and thus $z = x + y = 3$.
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