# A natural number multiplied by some integer results in a number with only ones and zeros

I recently solved a problem, which says that,

A positive integer can be multiplied with another integer resulting in
a positive integer that is composed only of one and zero as digits.

How can I prove that this is true(currently I assume that it is). Also, is it possible to establish an upper bound on the length(number of digits) of the number generated?

List all the numbers 1, 11, 111, … , 111…1 where the last one contains $n+1$ ones.
Look now at their remainders when divided by $n$. By the pigeonhole principle, two of them have the same remainder. But then their difference is of the form $1111..100000..0$ and is divisible by $n$..