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17s and 23s

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

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.
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