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17s and 23s

Stage: 3 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The two digit multiples of $17$ and $23$ are $17, 34, 51, 68, 85$ and $23, 46, 69, 92$ respectively. The second digit of a pair will be the first digit of the next pair so these pairs must follow on in a certain order, eg. $17$ must follow $51$ to give $517$.
There is only one pair that offers a choice of two options: $46$ can be followed by $68$ or $69$. Using $69$ forms a loop, $69, 92, 23, 34, 46$ (giving a repeated cycle of $5$ digits $69234$). Using $68$ leads to a dead end $68, 85, 52, 17 (68517). 17$ has no pair that can follow it.

To make a $2010$ digit number, there must be a large number of loops at the start. The number can either end on the loop (on any of the $5$ digits $6, 9, 2, 3, 4$) or it can end on the dead end (on any of the $4$ digits $8, 5, 1, 7$). This gives $9$ possible endings, so there must be $9$ such numbers.

This problem is taken from the UKMT Mathematical Challenges.