Find all such pairs a,b,c such that $a\equiv b \pmod c$, $b\equiv c \pmod a$,$c\equiv a \pmod b$

Question

This problem is taken from An Introduction to The Theory Of Numbers, by Ivan Niven.
Find all such pairs $a,b,c$ such that $a\equiv b \pmod c$, $b\equiv c \pmod a$, $c\equiv a \pmod b$. I wrote $a=b+ck_1,\ b=c+ak_2, c=a+bk_3$ and equated the determinant to $0$, but didn't get anything much useful out of it.

Answer 1

In order that the homogeneous system

$$\begin{cases}\ \ \ \ a&-&b&-&k_1c&=&0\\-k_2a&+&b&-&c&=&0\\ \ -a&-&k_3b&+&c&=&0\end{cases}$$

has a solution different from the trivial solution $(0,0,0)$, we must have:

$$\tag{1}\begin{vmatrix}1&-1&-k_1\\-k_2&1&-1\\-1&-k_3&1\end{vmatrix}=0 \ \ \iff \ \ k_1k_2k_3+k_1+k_2+k_3=0$$

Either one of the $k_i=0$ (giving for example when $k_1=0$, $k_2$ arbitrary and $k_3=-k_2$), or we can write $(1)$ under the form:

$$\tag{2}\iff \dfrac{1}{k_2k_3}+\dfrac{1}{k_3k_1}+\dfrac{1}{k_1k_2}=-1$$

and there are very few integers triples $(k_1,k_2,k_3)$ that verify $(2)$.

Up to you...