\[\begin{align} {\color{red}{6}} & \mid (6-1)!=5\times 4\times 3\times 2\times 1 \\ {\color{blue}{7}} & \not\mid (7-1)!=6\times 5\times 4\times 3\times 2\times 1 \\ {\color{red}{8}} & \mid (8-1)!=7\times 6\times 5\times 4\times 3\times 2\times 1 \\ {\color{red}{9}} & \mid (9-1)!=8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \\ {\color{red}{10}} & \mid (10-1)!=9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \\ {\color{blue}{11}} & \not\mid (11-1)!=10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \\ {\color{red}{12}} & \mid (12-1)!=11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \end{align}\]

The red numbers are the composite numbers (bigger than \(5\)), the blue numbers are the prime numbers (bigger than \(5\)).

Is it true, that for **any** composite positive integer \(k\) (\(>5\)),
\[{\color{red}{k}}\mid (k-1)!=(k-1)\times (k-2)\times \dots \times 1\]

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