Multiples of $17$ Multiples of $23$
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)
$69\rightarrow92\rightarrow23\rightarrow34$ 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.