Challenge Level

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.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.