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17s and 23s

Age 11 to 14 Short Challenge Level:

How many 2010-digit numbers are there in which every pair of consecutive digits forms a two digit number that is divisible by $17$ or $23$?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

 

 

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.