Prove or disprove the following:
For any integer number N there exists at least one integer number M, such that the decimal presentation of M*N needs only two distinct digits.
I assumed that 'only two distinct digits' meant exactly two for the very reason you mentioned, i.e. that if one digit is allowed then M=0 works for all N.
But if exactly two digits are required, then N=0 becomes an exception, which is why I noted it.