# Guess the numbers

One day Bob and Alice walk around they were having the following conversation:

Bob: Alice go ahead and pick any random number.

Alice: Okay, I have one.

Bob: Alice multiply the random number by X and sum every single digit(s) until it's Y digit(s) left.

Alice: Done.

Bob: I would bet that the result is an odd number.

Alice: Well, yeah. It's either odd or even either way. You just got lucky!

Bob: How about I said that it was X?

Alice: How do you know?

Bob: Well, that's simply because <reasoning>

What number are X and Y? Can anyone explain Bob's <reasoning>?

Fun fact: There is a related thread in Math.SE which explains Bob's reasoning. When the question digits are added it would be X-4.

Take an odd Y-digit number which equals to $$X$$. Let the original number be $$O$$. If we go through all the steps backwards, a bunch of odd or even numbers can be obtained. If $$X=9n+m$$, $$O(9n+m) = 9k+m$$, which means $$m$$ can only be 0 if it is to always work regardless of $$O$$. If even 2 as $$2X$$ ends up as $$X$$, in this case there can be only one step, and $$X=9$$, $$Y=1$$.