Solution May 22, 2007

Problem

Prove or disprove:

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

Solution

This is true.

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

$a_k = 7\cdot\frac{10^k-1}9$

modulo $n\,$ . This is an infinite sequence but there are only finitely many (namely, $n\,$ ) residues modulo $n\,$ so let $i<j\,$ be chosen such that $a_i = a_j \pmod n\,$ .

Then, the difference $a_j - a_i\,$ is divisible by $n\,$ . Its decimal digits form a sequence of 7s followed by 0s, so $m=\frac{a_j-a_i}n$ satisfies all requirements of the problem.

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