Problem E

Two Digits at a Time

The number \(34\,692\) contains \(5\) digits, and so we say its digit length is \(5\). The first digit of \(34\,692\) is \(3\). Also, the two-digit integers formed by choosing any pair of consecutive digits, that is, \(34\), \(46\), \(69\), and \(92\), are all divisible by either \(17\) or \(23\).

An integer with digit length \(2022\) has first digit \(3\). This integer also has the property that the two-digit integers formed by choosing any pair of consecutive digits are all divisible by either \(17\) or \(23\).

List all of the possibilities for the last three digits of this integer.