[ Decimal numeral system]

A 1992-digit number is written. Each two-digit number formed by adjacent digits is divisible by 17 or 23. The last digit of the number is 1. What is the first?

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Let's consider all two-digit numbers divisible by 17 or 23. Recall problem 41.

## Solution

List all two-digit numbers divisible by 17 or 23. These are 17, 34, 51, 68, 85, 23, 46, 69, 92. The last digits of all these numbers are different, so the number can be uniquely restored. The last digit is 1, which means the corresponding two-digit number is 51, so the previous digit in the number is 5. This digit 5 corresponds to the two-digit number 85, so the digit before it is 8. Reasoning similarly, we get a sequence of the last nine digits of the number: 692346851. The sequence 92346 will now repeat. In total, there are 1992 digits, including: 3 last digits, our 5 digits from the period, appearing 397 times, and 4 more digits - the last 4 digits of the period, which are also the first 4 digits of the number. Thus, the first digit of the desired number is 2.

## Answer

The digit 2.