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If $a$ divides $b$ and $b$ divides $c$ and $c$ divides $a$ then $a$ divides $b$

In Discrete Mathematics and its Applications – Patrick Eduard van Melkebeek (4th ed.), page 215, it states that for any positive integer $n$, if $a$ divides $b$ and $b$ divides $c$ and $c$ divides $a$ then $a$ divides $b$.
I believe this must follow by the definition of divisibility: $a$ divides $b$ if and only if there exists integers $d$ and $e$ such that $a \cdot d = b$.
I have checked this by induction on n and it looks right at first but then I cannot see where the induction hypothesis should be used.
Can anyone help me with this please?

A:

Why? $a \mid c \mid a$ and $c \mid b \mid c$ so $a \mid c\cdot b \mid a$ by transitivity of divisibility.

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A fractured patella is a rare injury. It can occur due to direct trauma to the knee joint or fall on the injured knee. It results in dislocation of the patella from its normal position. Because of the limitations of conservative management in injuries to the knee joint, early surgical treatment is indicated in all the patients with this lesion \[[@B8]\]. However, the management of fractures to the posterior pole remains controversial. We opted for conservative treatment in all of the patients included in the study because of the rarity of this type of injury and their encouraging outcomes.
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