Five. (25 points) Given positive integers $a$, $b$, $c$ satisfying $a < b < c$, and $ab + bc + ca = abc$. Find all sets of $a$, $b$, $c$ that meet the conditions.

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Given $1 \leqslant a<b<c$, we know $a b c=a b+b c+c a<3 b c$.
Therefore, $a<3$. Hence $a=1$ or $a=2$.
(1) When $a=1$, we have $b+b c+c=b c$, which simplifies to $b+c=0$, contradicting the fact that $b$ and $c$ are positive integers.
(2) When $a=2$, we have $2 b+b c+2 c=2 b c$, which simplifies to $b c-2 b-2 c=0$.
Thus, $(b-2)(c-2)=4$.
Also, since $2<b<c$, we have $0<b-2<c-2$.
Therefore, $b-2=1, c-2=4$.
Hence, $b=3, c=6$.
Therefore, the only set of positive integers that satisfy the conditions is:
$$
a=2, b=3, c=6 \text {. }
$$