17s and 23s

Answer: 9

Multiples of $17$         Multiples of $23$

        $17$                          $23$

        $34$                          $46$

        $51$                          $69$

        $68$                          $92$

        $85$

Can't use $17$ (except at the end) because none of the numbers start with $7$

Try starting from $34$:

                 $68\rightarrow85\rightarrow51\rightarrow17$   stuck (unless this is at the end)

             $\nearrow$

$34\rightarrow46$

             $\searrow$

                 $69\rightarrow92\rightarrow23\rightarrow34$   loop of all other numbers

The number will look like $...34692346923...$ and will go in loops or go into the 'stuck' sequence at the end

It could start with the $3,4,6,9$ or $2$ which gives $5$ options which are just loops.

It could go into the 'stuck' sequence at the end by going $8$ instead of $9$ after the last $6$. This works as long as $6$ isn't the last digit - so $4$ more options.

Total $9$ options.

This diagram shows how the numbers can be used.

Image