# Solution May 22, 2007

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### Problem

Prove or disprove:

For any positive integer there is a positive integer such that has only 0's and 7's as decimal digits.

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### Solution

This is true.

Consider the sequence 7, 77, 777, etc, formally

modulo . This is an infinite sequence but there are only finitely many (namely, ) residues modulo so let be chosen such that .

Then, the difference is divisible by . Its decimal digits form a sequence of 7s followed by 0s, so satisfies all requirements of the problem.