A cyclic system of congruences

Find all triples $(a,b,c)$ of positive integers such that the following congruences^FR are verified:
$\left\{\begin{array}{l} a \equiv b \pmod{c} \\ b \equiv c \pmod{a} \\ c \equiv a \pmod{b} \end{array}\right.$ where $a,b,c$ are positive integers.

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Solution by Sequoia

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Since the problem is symetric under any permutation of the variables $a,b,c$ , we can assume wlog that $0< c\leqslant b\leqslant a$
Then the second equation from the system can be written as follow:
$\exists k\geqslant0/ b=c+ka$ where $k$ is non-negative since $b\geqslant c$ and $a$ is a positive integer.

Now assume that $k\geqslant 1$ , then one gets $b=c+ka\geqslant c+a>a$ but this is absurd since $a\geqslant b$ . Hence $l=0$ and $b=c$ .

Then the first (and last) congruence can be rewritten as $b\mid a$ .

Reciprocally, any triple of the form $(bd, b, b)$ with $d,b$ positive integers, is a solution of this system.

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