Adjacent Factors

Answer: 9

Build up a long list
             $6$
          $2$ $6$ $3$
       $4$ $2$ $6$ $3$ $9$
    $8$ $4$ $2$ $6$ $3$ $9$ $1$
    $8$ $4$ $2$ $6$ $3$ $9$ $1$ $5$ $10$
All numbers except $7$ used

Can we also use $7$?
$7$ should go at the end because it can only go next to $1$
$7$ $1$ $?$
$5$ will only fit if $5$ comes next (using the $1$) or if $5$ comes last, after $10$, which would come after $2$

$5$ next:
$7$ $1$ $5$ $10$ $2$ $?$
If $8$ or $4$ used next, $3$, $6$, $9$ won't fit
If $6$ used next, $8$ and $4$ won't fit

$5$ last:
$7$ $1$ $...$ $?$ $2$ $10$ $5$
Exactly the same problem as before now happens before the $2$

We can't use the $7$. We can only use 9 numbers. 

The diagram below shows all of the possible connections that can be used.

 
There are many other sequences of nine numbers that follow the rule. Can you find them all?