Find all positive triples of positive integers a, b, c so that $\frac {a+1}{b}$ , $\frac {b+1}{c}$, $\frac {c+1}{a}$ are also integers. Find all positive triples of positive integers a, b, c so that $\frac {a+1}{b}$ , $\frac {b+1}{c}$, $\frac {c+1}{a}$ are also integers.  
WLOG, let a$\leqq b\leqq c$, 

If any two of $a,b,c$ are equal, then wlog. $a=b$. As $\frac{b+1}{a}=1+\frac1a$ is an integer, we conclude $a=b=1$. The remaining conditions are that $\frac{c+1}{1}$ and $\frac 2c$ are integers, which lead us to the solutions
$$(1,1,1),\qquad (1,1,2) $$
(and cyclic permutations of the latter).
So assume $a,b,c $ are pairwise different. By cyclic permutation, we may assume wlog that $a<b<c$ or that $a>b>c$.
In the first case, $0<\frac{a+1}{b}\le \frac bb=1$ and hence $a+1=b$. Likewise, $b+1=c$. Then the last integer is $\frac{c+1}a=\frac{a+3}a=1+\frac 3a$ and we must have $a=1$ or $a=3$, whic gives us the solutions
$$(1,2,3),\qquad (3,4,5) $$
(and cyclic permutations).
In the case $a>b>c$, we instead have that $0<\frac{c+1}{a}\le \frac{c+1}{c+2}<1$, not an integer. So this case does not produce additional solutions.
