Solution May 22, 2007


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.


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.