Can someone be so kind as to explain what is meant by the decimal expansion of an integer? I saw the following at this link but I don't know what decimal expansion of an integer refers to: https://madhavamathcompetition.com/2017/01/23/the-pigeon-hole-principle-some-notes-and-examples/ Example 4:
Show that for every integer $n$ there is a multiple of $n$ that has only $0$’s and $1$’s in its decimal expansion.
Let $n$ be a positive integer. Consider the $n+1$ integers $1, 11, 111, \ldots, 11\ldots 1$ (where the last integer in this list is the integer with $n+1$ $1$’s in its decimal expansion). Note that there are $n$ possible remainders when an integer is divided by $n$. Because there are $n+1$ integers in this list, by the pigeonhole principle, there must be two with the same remainder when divided by $n$. The larger of these integers less the smaller one is a multiple of $n$, which has a decimal expansion consisting entirely of $0$’s and $1$’s.
Perhaps I am misunderstanding the use of the word multiple. I thought it would mean an integer multiple.