# Problem of the Week 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.