You may also like

problem icon

Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

problem icon

Calendar Capers

Choose any three by three square of dates on a calendar page...

problem icon

Latin Numbers

Can you create a Latin Square from multiples of a six digit number?

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.